ALBERT

Description: We consider the problem of exponential utility indifference valuation under the simplified framework where traded and nontraded assets are uncorrelated but where the claim to be priced possibly depends on both. Traded asset prices follow a multivariate Black and Scholes model, while nontraded asset prices evolve as generalized Ornstein–Uhlenbeck processes. We provide a BSDE characterization of the utility indifference price (UIP) for a large class of non-smooth, possibly unbounded, payoffs depending simultaneously on both classes of assets. Focusing then on Vanilla claims and using the Gaussian structure of the model allows us to employ some BSDE techniques (in particular, a Malliavin-type representation theorem due to Ma and Zhang, Ann Appl Probab 12:1390–1418, 2002 ) to prove the regularity of Z and to characterize the UIP for possibly discontinuous Vanilla payoffs as a viscosity solution of a suitable PDE with continuous space derivatives. The optimal hedging strategy is also identified essentially as the delta hedging strategy corresponding to the UIP. Since there are no closed-form formulas in general, we also obtain asymptotic expansions for prices and hedging strategies when the risk aversion parameter is small. Finally, our results are applied to pricing and hedging power derivatives in various structural models for energy markets.

Description: This paper presents an alternative approach to the regularized least squares solution of ill-posed inverse problems. Instead of solving a minimization problem with an objective function composed of a data term and a regularization term, the regularization information is used to define a projection onto a convex subspace of regularized candidate solutions. The objective function is modified to include the projection of each iterate in the place of the regularization. Numerical experiments based on the problem of motion estimation for geophysical fluid images, show the improvement of the proposed method compared with regularization methods. For the presented test case, the incremental projection method uses 7 times less computation time than the regularization method, to reach the same error target. Moreover, at convergence, the incremental projection is two order of magnitude more accurate than the regularization method.

Description: We study a dynamical thin shallow shell whose elastic deformations are described by a nonlinear system of Marguerre–Vlasov’s type under the presence of thermal effects. Our main result is the proof of a global existence and uniqueness of a weak solution in the case of clamped boundary conditions. Standard techniques for uniqueness do not work directly in this case. We overcame this difficulty using recent work due to Lasiecka (Appl Anal 4:1376–1422, 1998 ).

Description: Time optimal control problems for systems with impulsive controls are investigated. Sufficient conditions for the existence of time optimal controls are given. A dynamical programming principle is derived and Lipschitz continuity of an appropriately defined value functional is established. The value functional satisfies a Hamilton–Jacobi–Bellman equation in the viscosity sense. A numerical example for a rider-swing system is presented and it is shown that the reachable set is enlargered by allowing for impulsive controls, when compared to nonimpulsive controls.

Description: We study the long-time behavior as time goes to infinity of global bounded solutions to the following nonautonomous semilinear viscoelastic equation: $$\begin{aligned} |u_t |^\rho u_{tt} -\Delta u_{tt}-\Delta u_{t}-\Delta u +\int ^\tau _0 k(s) \Delta u(t-s)ds+ f(x,u)=g, \ \tau \in \{t, \infty \}, \end{aligned}$$ in \({\mathbb {R}}^+\times \Omega \) , with Dirichlet boundary conditions, where \(\Omega \) is a bounded domain in \({\mathbb {R}}^n\) and the nonlinearity f is analytic. Based on an appropriate (perturbed) new Lyapunov function and the Łojasiewicz–Simon inequality we prove that any global bounded solution converges to a steady state. We discuss also the rate of convergence which is polynomial or exponential, depending on the Łojasiewicz exponent and the decay of the term g .

Description: This paper describes different representations for solution operators of Laplacian boundary value problems on bounded regions in \({\mathbb R}^N, N \ge 2\) and in exterior regions when \(N = 3\) . Null Dirichlet, Neumann and Robin boundary conditions are allowed and the results hold for weak solutions in relevant subspaces of Hilbert–Sobolev space associated with the problem. The solutions of these problems are shown to be strong limits of finite rank perturbations of the fundamental solution of the problem. For exterior regions these expressions generalize multipole expansions.

Description: We study a risk sensitive control version of the lifetime ruin probability problem. We consider a sequence of investments problems in Black–Scholes market that includes a risky asset and a riskless asset. We present a differential game that governs the limit behavior. We solve it explicitly and use it in order to find an asymptotically optimal policy.

Description: We study an optimal job-switching and consumption/investment problem of an infinitely-lived economic agent who exhibits constant relative risk aversion. We consider two kinds of jobs, one of which allows the agent to receive higher income but makes him suffer higher level of utility loss than the other. The job-switching opportunities are reversible in the sense that one can move from the current job to the other at any time. We provide the closed form solution for the optimal job-switching and consumption/investment policies by using the dynamic programming approach, and show various properties of the solution. We compare the optimal consumption/investment policies to those without job-switching opportunities. As a special case of our problem, we also compare the solution in the case where the agent has a reversible retirement option with that in the case where he has an irreversible retirement option.

Description: We consider the Navier–Stokes equation (N-S) in dimensions two and three as limits of the fractional approximations. In 2-D the N-S problem is critical with respect to the standard \(L^2\) a priori estimates and we consider its regular approximations with the fractional power operator \((-P\Delta )^{1+\alpha }\) , \(\alpha 〉0\) small, where P is the projector on the space of divergence-free functions. In 3-D different properties of the N-S problem with respect to the standard \(L^2\) a priori estimate are obtained and the 3-D regular approximating problem involves fractional power operator \((-P\Delta )^s\) with \(s〉\frac{5}{4}\) . Using Dan Henry’s semigroup approach and the Giga-Miyakawa estimates we construct regular solutions to such approximations. The solutions are global in time, unique, smooth and regularized through the equation in time. Solution to 2-D and 3-D N-S equations are obtained next as a limit of such regular solutions of the approximations. Moreover, since the nonlinearity of the N-S equation is of quadratic type , the solutions corresponding to small initial data and small f are shown to be global in time and regular.

Description: In this paper we consider space semi-discretization of some integro-differential equations using the harmonic analysis method. We study the problem of boundary observability, i. e., the problem of whether the initial data of solutions can be estimated uniformly in terms of the boundary observation as the net-spacing \(h\rightarrow 0\) . When \(h\rightarrow 0\) these finite-dimensional spaces increase and eventually cover the whole space. We thus recover the well-known observability property of the continuous system as the limit of discrete observability estimates as the mesh size tends to zero. We shall consider the piecewise Hermite cubic orthogonal spline collocation semi-discretization.

Description: We consider the problem of maximizing the expected amount of time an exponential martingale spends above a constant threshold up to a finite time horizon. We assume that at any time the volatility of the martingale can be chosen to take any value between \(\sigma _1\) and \(\sigma _2\) , where \(0 〈 \sigma _1 〈 \sigma _2\) . The optimal control consists in choosing the minimal volatility \(\sigma _1\) when the process is above the threshold, and the maximal volatility if it is below. We give a rigorous proof using classical verification and provide integral formulas for the maximal expected occupation time above the threshold.

Description: We consider an optimal control problem for the obstacle problem with an elliptic polyharmonic obstacle problem of order 2 m , where the obstacle function is assumed to be the control. We use a Moreau–Yosida approximate technique to introduce a family of problems governed by variational equations. Then, we prove optimal solutions existence and give an approximate optimality system and convergence results by passing to the limit in this system.

Description: The principal-agent problem in economics leads to variational problems subject to global constraints of b -convexity on the admissible functions, capturing the so-called incentive-compatibility constraints. Typical examples are minimization problems subject to a convexity constraint. In a recent pathbreaking article, Figalli et al. (J Econ Theory 146(2):454–478, 2011 ) identified conditions which ensure convexity of the principal-agent problem and thus raised hope on the development of numerical methods. We consider special instances of projections problems over b -convex functions and show how they can be solved numerically using Dykstra’s iterated projection algorithm to handle the b -convexity constraint in the framework of (Figalli et al. in J Econ Theory 146(2):454–478, 2011 ). Our method also turns out to be simple for convex envelope computations.

Description: We consider a two-point boundary value problem (TPBVP) in orbital mechanics involving a small body (e.g., a spacecraft or asteroid) and N larger bodies. The least action principle TPBVP formulation is converted into an initial value problem via the addition of an appropriate terminal cost to the action functional. The latter formulation is used to obtain a fundamental solution, which may be used to solve the TPBVP for a variety of boundary conditions within a certain class. In particular, the method of convex duality allows one to interpret the least action principle as a differential game, where an opposing player maximizes over an indexed set of quadratics to yield the gravitational potential. In the case where the time duration is less than a specific bound, there exists a unique critical point for the resulting differential game, which yields the fundamental solution given in terms of the solutions of associated Riccati equations.

Description: This paper is concerned with well-posedness and energy decay rates to a class of nonlinear viscoelastic Kirchhoff plates. The problem corresponds to a class of fourth order viscoelastic equations of \(p\) -Laplacian type which is not locally Lipschitz. The only damping effect is given by the memory component. We show that no additional damping is needed to obtain uniqueness in the presence of rotational forces. Then, we show that the general rates of energy decay are similar to ones given by the memory kernel, but generally not with the same speed, mainly when we consider the nonlinear problem with large initial data.

Description: We consider a well known model of a piezoelectric energy harvester. The harvester is designed as a beam with a piezoceramic layer attached to its top face (unimorph configuration). A pair of thin perfectly conductive electrodes is covering the top and the bottom faces of the piezoceramic layer. These electrodes are connected to a resistive load. The model is governed by a system consisting of two equations. The first of them is the equation of the Euler–Bernoulli model for the transverse vibrations of the beam and the second one represents the Kirchhoff’s law for the electric circuit. Both equations are coupled due to the direct and converse piezoelectric effects. The boundary conditions for the beam equations are of clamped-free type. We represent the system as a single operator evolution equation in a Hilbert space. The dynamics generator of this system is a non-selfadjoint operator with compact resolvent. Our main result is an explicit asymptotic formula for the eigenvalues of this generator, i.e., we perform the modal analysis for electrically loaded (not short-circuit) system. We show that the spectrum splits into an infinite sequence of stable eigenvalues that approaches a vertical line in the left half plane and possibly of a finite number of unstable eigenvalues. This paper is the first in a series of three works. In the second one we will prove that the generalized eigenvectors of the dynamics generator form a Riesz basis (and, moreover, a Bari basis) in the energy space. In the third paper we will apply the results of the first two to control problems for this model.

Description: We give a survey of recent results on flow-structure interactions modeled by a modified wave equation coupled at an interface with equations of nonlinear elasticity. Both subsonic and supersonic flow velocities are considered. The focus of the discussion here is on the interesting mathematical aspects of physical phenomena occurring in aeroelasticity, such as flutter and divergence. This leads to a partial differential equation treatment of issues such as well-posedness of finite energy solutions, and long-time (asymptotic) behavior. The latter includes theory of asymptotic stability, convergence to equilibria, and to global attracting sets. We complete the discussion with several well known observations and conjectures based on experimental/numerical studies.

Description: We consider configurational variations of a homogeneous (anisotropic) linear elastic material \(\Omega \subset \mathbb {R}^n\) with a crack K . First, we provide a simple way to compute configurational variations of energy by means of a volume integral. Then, under increasing information on the regularity of the displacement field we show how to obtain classical representations of the energy release due to Eshelby, Rice and Irwin. A rigorous functional setting for these representations to hold is provided.

Description: We analyze some systems of partial differential equations arising in the theory of mean field type control with congestion effects. We look for weak solutions. Our main result is the existence and uniqueness of suitably defined weak solutions, which are characterized as the optima of two optimal control problems in duality.

Description: 〈h3〉Abstract〈/h3〉 〈p〉We construct an abstract framework in which the dynamic programming principle (DPP) can be readily proven. It encompasses a broad range of common stochastic control problems in the weak formulation, and deals with problems in the “martingale formulation” with particular ease. We give two illustrations; first, we establish the DPP for general controlled diffusions and show that their value functions are viscosity solutions of the associated Hamilton–Jacobi–Bellman equations under minimal conditions. After that, we show how to treat singular control on the example of the classical monotone-follower problem. 〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We consider optimization problems on manifolds with equality and inequality constraints. A large body of work treats constrained optimization in Euclidean spaces. In this work, we consider extensions of existing algorithms from the Euclidean case to the Riemannian case. Thus, the variable lives on a known smooth manifold and is further constrained. In doing so, we exploit the growing literature on unconstrained Riemannian optimization. For the special case where the manifold is itself described by equality constraints, one could in principle treat the whole problem as a constrained problem in a Euclidean space. The main hypothesis we test here is whether it is sometimes better to exploit the geometry of the constraints, even if only for a subset of them. Specifically, this paper extends an augmented Lagrangian method and smoothed versions of an exact penalty method to the Riemannian case, together with some fundamental convergence results. Numerical experiments indicate some gains in computational efficiency and accuracy in some regimes for minimum balanced cut, non-negative PCA and 〈em〉k〈/em〉-means, especially in high dimensions.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉This paper associates a dual problem to the minimization of an arbitrary linear perturbation of the robust sum function introduced in Dinh et al. (Set Valued Var Anal, 2019). It provides an existence theorem for primal optimal solutions and, under suitable duality assumptions, characterizations of the primal–dual optimal set, the primal optimal set, and the dual optimal set, as well as a formula for the subdifferential of the robust sum function. The mentioned results are applied to get simple formulas for the robust sums of subaffine functions (a class of functions which contains the affine ones) and to obtain conditions guaranteeing the existence of best approximate solutions to inconsistent convex inequality systems.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The aim of the present work is to provide an explicit decomposition formula for the resolvent operator 〈span〉 〈span〉\(\mathrm {J}_{A+B}\)〈/span〉 〈/span〉 of the sum of two set-valued maps 〈em〉A〈/em〉 and 〈em〉B〈/em〉 in a Hilbert space. For this purpose we introduce a new operator, called the 〈em〉A〈/em〉-resolvent operator of 〈em〉B〈/em〉 and denoted by 〈span〉 〈span〉\(\mathrm {J}^A_B\)〈/span〉 〈/span〉, which generalizes the usual notion. Then, our main result lies in the decomposition formula 〈span〉 〈span〉\(\mathrm {J}_{A+B}=\mathrm {J}_A\circ \mathrm {J}^A_B\)〈/span〉 〈/span〉 holding true when 〈em〉A〈/em〉 is monotone. Several properties of 〈span〉 〈span〉\(\mathrm {J}^A_B\)〈/span〉 〈/span〉 are deeply investigated in this paper. In particular the relationship between 〈span〉 〈span〉\(\mathrm {J}^A_B\)〈/span〉 〈/span〉 and an extended version of the classical Douglas–Rachford operator is established, which allows us to propose a weakly convergent algorithm that computes numerically 〈span〉 〈span〉\(\mathrm {J}^A_B\)〈/span〉 〈/span〉 (and thus 〈span〉 〈span〉\(\mathrm {J}_{A+B}\)〈/span〉 〈/span〉 from the decomposition formula) when 〈em〉A〈/em〉 and 〈em〉B〈/em〉 are maximal monotone. In order to illustrate our theoretical results, we give an application in elliptic PDEs. Precisely the decomposition formula is used to point out the relationship between the classical obstacle problem and a new nonlinear PDE involving a partially blinded elliptic operator. Some numerical experiments, using the finite element method, are carried out in order to support our approach.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We extend classical results on variational inequalities with convex sets with gradient constraint to a new class of fractional partial differential equations in a bounded domain with constraint on the distributional Riesz fractional gradient, the 〈span〉 〈span〉\(\sigma \)〈/span〉 〈/span〉-gradient (〈span〉 〈span〉\(0〈\sigma 〈1\)〈/span〉 〈/span〉). We establish continuous dependence results with respect to the data, including the threshold of the fractional 〈span〉 〈span〉\(\sigma \)〈/span〉 〈/span〉-gradient. Using these properties we give new results on the existence to a class of quasi-variational variational inequalities with fractional gradient constraint via compactness and via contraction arguments. Using the approximation of the solutions with a family of quasilinear penalisation problems we show the existence of generalised Lagrange multipliers for the 〈span〉 〈span〉\(\sigma \)〈/span〉 〈/span〉-gradient constrained problem, extending previous results for the classical gradient case, i.e., with 〈span〉 〈span〉\(\sigma =1\)〈/span〉 〈/span〉.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this work, we propose a new algorithm for finding a zero of the sum of two monotone operators where one is assumed to be single-valued and Lipschitz continuous. This algorithm naturally arises from a non-standard discretization of a continuous dynamical system associated with the Douglas–Rachford splitting algorithm. More precisely, it is obtained by performing an explicit, rather than implicit, discretization with respect to one of the operators involved. Each iteration of the proposed algorithm requires the evaluation of one forward and one backward operator.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We study here a singular perturbation problem of biLaplacian type, which can be seen as the biharmonic counterpart of classical combustion models. We provide different results, that include the convergence to a free boundary problem driven by a biharmonic operator, as introduced in Dipierro et al. (〈a href="http://arxiv.org/abs/1808.07696"〉arXiv:1808.07696〈/a〉, 〈span〉2018〈/span〉), and a monotonicity formula in the plane. For the latter result, an important tool is provided by an integral identity that is satisfied by solutions of the singular perturbation problem. We also investigate the quadratic behaviour of solutions near the zero level set, at least for small values of the perturbation parameter. Some counterexamples to the uniform regularity are also provided if one does not impose some structural assumptions on the forcing term. 〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We consider a two player, zero sum differential game with a cost of Bolza type, subject to a state constraint. It is shown that, under a suitable hypothesis concerning existence of inward pointing velocity vectors for the minimizing player at the boundary of the constraint set, the lower value of the game is Lipschitz continuous and is the unique viscosity solution (appropriately defined) of the lower Hamilton-Jacobi-Isaacs equation. If the inward pointing hypothesis is satisfied by the maximizing player’s velocity set, then the upper game is Lipschitz continuous and is the unique solution of the upper Hamilton-Jacobi-Isaacs equation. Under the classical Isaacs condition, the upper and lower Hamilton-Jacobi-Isaacs equation coincide. In this case, even if the inward pointing hypothesis is satisfied w.r.t. both players, the value of the game might fail to exist; however imposing stronger constraint qualifications (involving the existence of inward pointing vectors associated with saddle points for the Hamiltonian) the game value does exist and is the unique solution to this Hamilton-Jacobi-Isaacs equation. The novelty of our work resides in the fact that we permit the two players’ controls to be completely coupled within the dynamic constraint, state constraint and the cost functional, in contrast to earlier work, in which the players’ controls are decoupled w.r.t. the dynamics and state constraint, and interaction between them only occurs through the cost function. Furthermore, the inward pointing hypotheses that we impose are of a verifiable nature and less restrictive than those earlier employed.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this paper we consider non-smooth convex optimization problems with (possibly) infinite intersection of constraints. In contrast to the classical approach, where the constraints are usually represented as intersection of simple sets, which are easy to project onto, in this paper we consider that each constraint set is given as the level set of a convex but not necessarily differentiable function. For these settings we propose subgradient iterative algorithms with random minibatch feasibility updates. At each iteration, our algorithms take a subgradient step aimed at only minimizing the objective function and then a subsequent subgradient step minimizing the feasibility violation of the observed minibatch of constraints. The feasibility updates are performed based on either parallel or sequential random observations of several constraint components. We analyze the convergence behavior of the proposed algorithms for the case when the objective function is strongly convex and with bounded subgradients, while the functional constraints are endowed with a bounded first-order black-box oracle. For a diminishing stepsize, we prove sublinear convergence rates for the expected distances of the weighted averages of the iterates from the constraint set, as well as for the expected suboptimality of the function values along the weighted averages. Our convergence rates are known to be optimal for subgradient methods on this class of problems. Moreover, the rates depend explicitly on the minibatch size and show when minibatching helps a subgradient scheme with random feasibility updates.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The equality between dissipation and energy drop is a structural property of gradient-flow dynamics. The classical implicit Euler scheme fails to reproduce this equality at the discrete level. We discuss two modifications of the Euler scheme satisfying an exact energy equality at the discrete level. Existence of discrete solutions and their convergence as the fineness of the partition goes to zero are discussed. Eventually, we address extensions to generalized gradient flows, GENERIC flows, and curves of maximal slope in metric spaces. 〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We consider a local minimizer, in the sense of the 〈span〉 〈span〉\(W^{1,m}\)〈/span〉 〈/span〉 norm (〈span〉 〈span〉\(m\ge 1\)〈/span〉 〈/span〉), of the classical problem of the calculus of variations 〈span〉 〈equationnumber〉P〈/equationnumber〉 〈span〉$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathrm{Minimize}}\quad &{}\displaystyle I(x):=\int _a^b\varLambda (t,x(t), x'(t))\,dt+\varPsi (x(a), x(b))\\ \text {subject to:} &{}x\in W^{1,m}([a,b];\mathbb {R}^n),\\ &{}x'(t)\in C\,\text { a.e., } \,x(t)\in \varSigma \quad \forall t\in [a,b].\\ \end{array}\right. } \end{aligned}$$〈/span〉 〈/span〉where 〈span〉 〈span〉\(\varLambda :[a,b]\times \mathbb {R}^n\times \mathbb {R}^n\rightarrow \mathbb {R}\cup \{+\infty \}\)〈/span〉 〈/span〉 is just Borel measurable, 〈em〉C〈/em〉 is a cone, 〈span〉 〈span〉\(\varSigma \)〈/span〉 〈/span〉 is a nonempty subset of 〈span〉 〈span〉\(\mathbb {R}^n\)〈/span〉 〈/span〉 and 〈span〉 〈span〉\(\varPsi \)〈/span〉 〈/span〉 is an arbitrary possibly extended valued function. When 〈span〉 〈span〉\(\varLambda \)〈/span〉 〈/span〉 is real valued, we merely assume a local Lipschitz condition on 〈span〉 〈span〉\(\varLambda \)〈/span〉 〈/span〉 with respect to 〈em〉t〈/em〉, allowing 〈span〉 〈span〉\(\varLambda (t,x,\xi )\)〈/span〉 〈/span〉 to be discontinuous and nonconvex in 〈em〉x〈/em〉 or 〈span〉 〈span〉\(\xi \)〈/span〉 〈/span〉. In the case of an extended valued Lagrangian, we impose the lower semicontinuity of 〈span〉 〈span〉\(\varLambda (\cdot ,x,\cdot )\)〈/span〉 〈/span〉, and a condition on the effective domain of 〈span〉 〈span〉\(\varLambda (t,x,\cdot )\)〈/span〉 〈/span〉. We use a recent variational Weierstrass type inequality to show that the minimizers satisfy a relaxation result and an Erdmann – Du Bois-Reymond convex inclusion which, remarkably, holds whenever 〈span〉 〈span〉\(\varLambda (x,\xi )\)〈/span〉 〈/span〉 is autonomous and just Borel. Under a growth condition, weaker than superlinearity, we infer the Lipschitz continuity of minimizers.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The main goal of this paper is to study the existence of solutions for optimal control problems governed by mixed quasi-equilibrium problems under monotonicity type conditions. More precisely, the state control system takes the general form of a mixed quasi-equilibrium problem described by the sum of a maximal monotone bifunction and a pseudomonotone bifunction in the sense of Brézis. As applications, we study the existence of solutions for optimal control problems governed by quasi-variational inequalities. In particularly, we consider the optimal control of an evolutionary quasi-variational inequality described by a 〈em〉p〈/em〉-Laplacian type operator. The results obtained in this paper are new and can be applied to the study of optimal control of a variety of systems whose formulations can be presented as a mixed equilibrium problem.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this paper, we deal with the existence of global mild solutions and asymptotic behavior to the viscous Camassa–Holm equation in the locally uniform spaces. First we establish the global well-posedness for the Cauchy problem of viscous Camassa–Holm equation in 〈span〉 〈span〉\({\mathbb {R}}^1\)〈/span〉 〈/span〉 for any initial data 〈span〉 〈span〉\(u_0\in {\dot{H}}^1_U({\mathbb {R}}^1).\)〈/span〉 〈/span〉 Then we study the long time dynamical behavior of non-autonomous viscous Camassa–Holm equation on 〈span〉 〈span〉\({\mathbb {R}}^1\)〈/span〉 〈/span〉 with 〈em〉a new class of external forces〈/em〉 and show the existence of 〈span〉 〈span〉\((H^1_U({\mathbb {R}}^1),H^1_\phi ({\mathbb {R}}^1))\)〈/span〉 〈/span〉-uniform(w.r.t. 〈span〉 〈span〉\(g\in \mathcal {H}_{L^2_U({\mathbb {R}}^1)}(g_0)\)〈/span〉 〈/span〉) attractor 〈span〉 〈span〉\(\mathcal {A}_{\mathcal {H}_{L^2_U({\mathbb {R}}^1)}(g_0)}\)〈/span〉 〈/span〉 with locally uniform external forces being translation uniform bounded but not translation compact in 〈span〉 〈span〉\(L_b^2({\mathbb {R}};L^2_U({\mathbb {R}}^1))\)〈/span〉 〈/span〉.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉A distributed optimal control problem for an extended model of phase field type for tumor growth is addressed. In this model, the chemotaxis effects are also taken into account. The control is realized by two control variables that design the dispensation of some drugs to the patient. The cost functional is of tracking type, whereas the potential setting has been kept quite general in order to allow regular and singular potentials to be considered. In this direction, some relaxation terms have been introduced in the system. We show the well-posedness of the state system, the Fréchet differentiability of the control-to-state operator in a suitable functional analytic framework, and, lastly, we characterize the first-order necessary conditions of optimality in terms of a variational inequality involving the adjoint variables.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The Baillon–Haddad theorem establishes that the gradient of a convex and continuously differentiable function defined in a Hilbert space is 〈span〉 〈span〉\(\beta \)〈/span〉 〈/span〉-Lipschitz if and only if it is 〈span〉 〈span〉\(1/\beta \)〈/span〉 〈/span〉-cocoercive. In this paper, we extend this theorem to Gâteaux differentiable and lower semicontinuous convex functions defined on an open convex set of a Hilbert space. Finally, we give a characterization of 〈span〉 〈span〉\(C^{1,+}\)〈/span〉 〈/span〉 convex functions in terms of local cocoercivity.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We study a nonlinear system of partial differential equations arising in macroeconomics which utilizes a mean field approximation. This system together with the corresponding data, subject to two moment constraints, is a model for debt and wealth across a large number of similar households, and was introduced in a recent paper of Achdou et al. (Philos Trans R Soc Lond Ser A 372(2028):20130397, 2014). We introduce a relaxation of their problem, generalizing one of the moment constraints; any solution of the original model is a solution of this relaxed problem. We prove existence and uniqueness of strong solutions to the relaxed problem, under the assumption that the time horizon is small. Since these solutions are unique and since solutions of the original problem are also solutions of the relaxed problem, we conclude that if the original problem does have solutions, then such solutions must be the solutions we prove to exist. Furthermore, for some data and for sufficiently small time horizons, we are able to show that solutions of the relaxed problem are in fact not solutions of the original problem. In this way we demonstrate nonexistence of solutions for the original problem in certain cases.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this paper, we investigate the asymptotic stability of global solutions to the following intermittently controlled semilinear viscoelastic equation with memory term 〈span〉 〈span〉$$\begin{aligned}&u_{tt}-\varDelta u_{tt}-\varDelta u+\int _{0}^{\tau }k(s)\varDelta u(t-s)ds+h_{1}(t)u_{t} -h_{2}(t)\varDelta u_{t}\\&\quad =f(u),\;(x,t)\in \varOmega \times [0,\infty ), \end{aligned}$$〈/span〉 〈/span〉under the null Dirichlet boundary condition and 〈span〉 〈span〉\(\tau \in \{t,\infty \}\)〈/span〉 〈/span〉. By virtue of appropriate new Lyapunov functional and Łojasiewicz–Simon inequality, we show that any global bounded solution converges to a steady state and get the rate of convergence as well, when damping coefficients are integral positive and positive-negative, respectively. Moreover, under the assumptions of on–off or sign-changing dampings, we derive the asymptotic stability of solutions.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We analyze two 〈span〉 〈span〉\(H^{-1}\)〈/span〉 〈/span〉-least-squares methods for the steady Navier–Stokes system of incompressible viscous fluids. Precisely, we show the convergence of minimizing sequences for the least-squares functional toward solutions. Numerical experiments support our analysis.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this paper, we consider a linear one-dimensional thermoelastic Bresse system with second sound consisting of three hyperbolic equations and two parabolic equations coupled in a certain manner under mixed homogeneous Dirichlet–Neumann boundary conditions, where the heat conduction is given by Cattaneo’s law. Only the longitudinal displacement is damped via the dissipation from the two parabolic equations, and the vertical displacement and shear angle displacement are free. We prove the well-posedness of the system and some exponential, non exponential and polynomial stability results depending on the coefficients of the equations and the smoothness of initial data. Our method of proof is based on the semigroup theory and a combination of the energy method and the frequency domain approach.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉This paper is concerned with the differential sensitivity analysis of variational inequalities in Banach spaces whose solution operators satisfy a generalized Lipschitz condition. We prove a sufficient criterion for the directional differentiability of the solution map that turns out to be also necessary for elliptic variational inequalities in Hilbert spaces (even in the presence of asymmetric bilinear forms, nonlinear operators and nonconvex functionals). Our method of proof is fully elementary. Moreover, our technique allows us to also study those cases where the variational inequality at hand is not uniquely solvable and where directional differentiability can only be obtained w.r.t. the weak or the weak-star topology of the underlying space. As tangible examples, we consider a variational inequality arising in elastoplasticity, the projection onto prox-regular sets, and a bang–bang optimal control problem.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this paper, we study an optimal control problem for a two-dimensional Cahn–Hilliard–Darcy system with mass sources that arises in the modeling of tumor growth. The aim is to monitor the tumor fraction in a finite time interval in such a way that both the tumor fraction, measured in terms of a tracking type cost functional, is kept under control and minimal harm is inflicted to the patient by administering the control, which could either be a drug or nutrition. We first prove that the optimal control problem admits a solution. Then we show that the control-to-state operator is Fréchet differentiable between suitable Banach spaces and derive the first-order necessary optimality conditions in terms of the adjoint variables and the usual variational inequality.〈/p〉

Description: In this paper, we consider the continuous-time nonzero-sum constrained stochastic games with the discounted cost criteria. The state space is denumerable and the action space of each player is a general Polish space, while the transition rates and cost functions are allowed to be unbounded from below and from above. The strategies for each player may be history-dependent and randomized. Models with these features seemingly have not been handled in the previous literature. By constructing a sequence of continuous-time finite-state game models to approximate the original denumerable-state game model, we prove the existence of constrained Nash equilibria for the constrained games with denumerable states.

Description: This paper is devoted to the study of the large time behaviour of viscosity solutions of parabolic equations with Neumann boundary conditions. This work is the sequel of Hu et al. (SIAM J Control Optim 53:378–398, 2015 ) in which a probabilistic method was developed to show that the solution of a parabolic semilinear PDE behaves like a linear term \(\lambda T\) shifted with a function v , where \((v,\lambda )\) is the solution of the ergodic PDE associated to the parabolic PDE. We adapt this method in finite dimension by a penalization method in order to be able to apply an important basic coupling estimate result and with the help of a regularization procedure in order to avoid the lack of regularity of the coefficients in finite dimension. The advantage of our method is that it gives an explicit rate of convergence.

Description: We discuss the parallel between the third-order Moore–Gibson–Thompson equation $$\begin{aligned} {\partial _{ttt}} u + \alpha {\partial _{tt}}u-\beta \Delta {\partial _t} u - \gamma \Delta u =0 \end{aligned}$$ depending on the parameters \(\alpha ,\beta ,\gamma 〉0,\) and the equation of linear viscoelasticity $$\begin{aligned} \partial _{tt}u(t) - \kappa (0)\Delta u(t) - \int _{0}^\infty \kappa ^{\prime }(s)\Delta u(t-s)\,\mathrm{d}s=0 \end{aligned}$$ for the particular choice of the exponential kernel $$\begin{aligned} \kappa (s) = a \mathrm{e}^{-b s} + c \end{aligned}$$ with \(a,b,c〉0\) . In particular, the latter model is shown to exhibit a preservation of regularity for a certain class of initial data, which is unexpected in presence of a general memory kernel \(\kappa \) .

Description: We introduce a dynamic optimization framework in which collateral is used to mitigate losses arising at counterparty’s default. The investor faces two sources of risk: the default risk of the entity referencing the traded credit swap security, and counterparty risk generated from the default event of the trading counterparty. We show that the value function of the control problem coincides with the classical solution of a nonlinear dynamic programming equation. We provide an explicit characterization of the optimal investment strategy, and show that the investor does not trade if counterparty risk is sufficiently high. These findings suggest that moving credit swap trades into well-designed clearinghouses may stimulate economic activities.

Description: We study a system of partial differential equations used to describe Bertrand and Cournot competition among a continuum of producers of an exhaustible resource. By deriving new a priori estimates, we prove the existence of classical solutions under general assumptions on the data. Moreover, under an additional hypothesis we prove uniqueness.

Description: Our aim in this paper is to study the well-posedness and the dissipativity of higher-order anisotropic conservative phase-field systems. More precisely, we prove the existence and uniqueness of solutions and the existence of the global attractor.

Description: The problem of identifying the diffusion parameter appearing in a nonlocal steady diffusion equation is considered. The identification problem is formulated as an optimal control problem having a matching functional as the objective of the control and the parameter function as the control variable. The analysis makes use of a nonlocal vector calculus that allows one to define a variational formulation of the nonlocal problem. In a manner analogous to the local partial differential equations counterpart, we demonstrate, for certain kernel functions, the existence of at least one optimal solution in the space of admissible parameters. We introduce a Galerkin finite element discretization of the optimal control problem and derive a priori error estimates for the approximate state and control variables. Using one-dimensional numerical experiments, we illustrate the theoretical results and show that by using nonlocal models it is possible to estimate non-smooth and discontinuous diffusion parameters.

Description: Given an open bounded smooth set \(\Omega \) in \(\mathrm{I}\!\mathrm{R}^N,\ N\geqslant 3\) , we provide a sufficient condition on the data \(f\) integrable with respect to the distance \(\delta \) , to ensure the blow-up of the gradient of the very weak solution for the Dirichlet equation.

Description: A boundary control problem for the viscous Cahn–Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first order necessary conditions for optimality are proved.

Description: A finite horizon optimal stopping problem for an infinite dimensional diffusion X is analyzed by means of variational techniques. The diffusion is driven by a SDE on a Hilbert space \(\mathcal {H}\) with a non-linear diffusion coefficient \(\sigma (X)\) and a generic unbounded operator A in the drift term. When the gain function \(\Theta \) is time-dependent and fulfils mild regularity assumptions, the value function \(\mathcal {U}\) of the optimal stopping problem is shown to solve an infinite-dimensional, parabolic, degenerate variational inequality on an unbounded domain. Once the coefficient \(\sigma (X)\) is specified, the solution of the variational problem is found in a suitable Banach space \(\mathcal {V}\) fully characterized in terms of a Gaussian measure \(\mu \) . This work provides the infinite-dimensional counterpart, in the spirit of Bensoussan and Lions (Application of variational inequalities in stochastic control, 1982 ), of well-known results on optimal stopping theory and variational inequalities in \(\mathbb {R}^n\) . These results may be useful in several fields, as in mathematical finance when pricing American options in the HJM model.

Description: In the present paper we study the convergence of the solution of the two dimensional (2-D) stochastic Leray- \(\alpha \) model to the solution of the 2-D stochastic Navier–Stokes equations. We are mainly interested in the rate, as \(\alpha \rightarrow 0\) , of the following error function $$\begin{aligned} \varepsilon _\alpha (t)=\sup _{s\in [0,t]} |\mathbf {u}^\alpha (s)-\mathbf {u}(s)|+\left( \int _0^t |\mathrm {A}^\frac{1}{2}[\mathbf {u}^\alpha (s)-\mathbf {u}(s)] |^2 ds \right) ^\frac{1}{2}, \end{aligned}$$ where \(\mathbf {u}^\alpha \) and \(\mathbf {u}\) are the solution of stochastic Leray- \(\alpha \) model and the stochastic Navier–Stokes equations, respectively. We show that when properly localized the error function \(\varepsilon _\alpha \) converges in mean square as \(\alpha \rightarrow 0\) and the convergence is of order \(O(\alpha )\) . We also prove that \(\varepsilon _\alpha \) converges in probability to zero with order at most \(O(\alpha )\) .

Description: We perform a numerical optimisation of the low frequencies of the Dirichlet Laplacian with perimeter and surface area restrictions, in two and 3-dimensions, respectively. In the former case, we handle the first 50 eigenvalues and measure the rate at which the corresponding optimisers approach the disk, while in the latter we optimise the first twenty eigenvalues. We derive theoretical compatibility conditions which must be satisfied by a sequence of optimisers and test our numerical results against these. We also consider the cases of rectangles with a fixed perimeter and parallelepipeds with a surface restriction for which we compute the first \(10^7\) and \(10^6\) optimal eigenvalues, respectively. In this context, we prove convergence to the cube in any dimensions and compare the numerical results with our theoretical estimates for the rate of convergence.

Description: We prove an existence and uniqueness result for a class of subdifferential inclusions which involve a history-dependent operator. Then we specialize this result in the study of a class of history-dependent hemivariational inequalities. Problems of such kind arise in a large number of mathematical models which describe quasistatic processes of contact. To provide an example we consider an elastic beam in contact with a reactive obstacle. The contact is modeled with a new and nonstandard condition which involves both the subdifferential of a nonconvex and nonsmooth function and a Volterra-type integral term. We derive a variational formulation of the problem which is in the form of a history-dependent hemivariational inequality for the displacement field. Then, we use our abstract result to prove its unique weak solvability. Finally, we consider a numerical approximation of the model, solve effectively the approximate problems and provide numerical simulations.

Description: In the paper we consider the problem of valuation of American options written on dividend-paying assets whose price dynamics follow the classical multidimensional Black and Scholes model. We provide a general early exercise premium representation formula for options with payoff functions which are convex or satisfy mild regularity assumptions. Examples include index options, spread options, call on max options, put on min options, multiply strike options and power-product options. In the proof of the formula we exploit close connections between the optimal stopping problems associated with valuation of American options, obstacle problems and reflected backward stochastic differential equations.

Description: In our paper we consider an infinite horizon consumption-investment problem under a model misspecification in a general stochastic factor model. We formulate the problem as a stochastic game and finally characterize the saddle point and the value function of that game using an ODE of semilinear type, for which we provide a proof of an existence and uniqueness theorem for its solution. Such equation is interested on its own right, since it generalizes many other equations arising in various infinite horizon optimization problems.

Description: We study a Stackelberg strategy subject to the evolutionary linearized micropolar fluids equations in domains with moving boundaries, considering a Nash multi-objective equilibrium (non necessarily cooperative) for the “follower players” (as is called in the economy field) and an optimal problem for the leader player with approximate controllability objective. We will obtain the following main results: the existence and uniqueness of Nash equilibrium and its characterization, the approximate controllability of the linearized micropolar system with respect to the leader control and the existence and uniqueness of the Stackelberg–Nash problem, where the optimality system for the leader is given.

Description: This work deals with the homogenization of functionals with linear growth in the context of \(\mathcal A\) -quasiconvexity. A representation theorem is proved, where the new integrand function is obtained by solving a cell problem where the coupling between homogenization and the \(\mathcal A\) -free condition plays a crucial role. This result extends some previous work to the linear case, thus allowing for concentration effects.

Description: The nonclassical diffusion equation with hereditary memory $$\begin{aligned} u_t-\Delta u_t-\Delta u-\int _0^\infty \kappa (s)\Delta u(t-s)\,\mathrm{d}s +\varphi (u)=f \end{aligned}$$ on a 3D bounded domain is considered, for a very general class of memory kernels \(\kappa \) . Setting the problem both in the classical past history framework and in the more recent minimal state one, the related solution semigroups are shown to possess finite-dimensional regular exponential attractors.

Description: In this paper we study two-person nonzero-sum games for continuous-time jump processes with the randomized history-dependent strategies under the finite-horizon payoff criterion. The state space is countable, and the transition rates and payoff functions are allowed to be unbounded from above and from below. Under the suitable conditions, we introduce a new topology for the set of all randomized Markov multi-strategies and establish its compactness and metrizability. Then by constructing the approximating sequences of the transition rates and payoff functions, we show that the optimal value function for each player is a unique solution to the corresponding optimality equation and obtain the existence of a randomized Markov Nash equilibrium. Furthermore, we illustrate the applications of our main results with a controlled birth and death system.

Description: In this paper we study the optimal transport of projected semiclassical measures on the flat torus which are linked to a class of Schrödinger’s type equations.

Description: We consider parametric equations driven by the sum of a p -Laplacian and a Laplace operator (the so-called ( p , 2)-equations). We study the existence and multiplicity of solutions when the parameter \(\lambda 〉0\) is near the principal eigenvalue \(\hat{\lambda }_1(p)〉0\) of \((-\Delta _p,W^{1,p}_{0}(\Omega ))\) . We prove multiplicity results with precise sign information when the near resonance occurs from above and from below of \(\hat{\lambda }_1(p)〉0\) .

Description: A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities: The optimal terminal wealth \(X^*(T) : = X_{\varphi ^*}(T)\) of the problem to maximize the expected U -utility of the terminal wealth \(X_{\varphi }(T)\) generated by admissible portfolios \(\varphi (t); 0 \le t \le T\) in a market with the risky asset price process modeled as a semimartingale; The optimal scenario \(\frac{dQ^*}{dP}\) of the dual problem to minimize the expected V -value of \(\frac{dQ}{dP}\) over a family of equivalent local martingale measures Q , where V is the convex conjugate function of the concave function U . In this paper we consider markets modeled by Itô-Lévy processes. In the first part we use the maximum principle in stochastic control theory to extend the above relation to a dynamic relation, valid for all \(t \in [0,T]\) . We prove in particular that the optimal adjoint process for the primal problem coincides with the optimal density process, and that the optimal adjoint process for the dual problem coincides with the optimal wealth process; \(0 \le t \le T\) . In the terminal time case \(t=T\) we recover the classical duality connection above. We get moreover an explicit relation between the optimal portfolio \(\varphi ^*\) and the optimal measure \(Q^*\) . We also obtain that the existence of an optimal scenario is equivalent to the replicability of a related T -claim. In the second part we present robust (model uncertainty) versions of the optimization problems in (i) and (ii), and we prove a similar dynamic relation between them. In particular, we show how to get from the solution of one of the problems to the other. We illustrate the results with explicit examples.

Description: We consider an abstract Cauchy problem for a certain class of linear differential equations in Hilbert space. We obtain a criterion for stability on some dense subsets of the state space of the \(C_0\) -semigroups in terms of location of eigenvalues of their infinitesimal generators (so-called polynomial stability). We apply this result to analysis of stability and stabilizability of special class of neutral type systems with distributed delay.

Description: In this paper, we investigate a periodic food chain model with harvesting, where the predators have size structures and are described by first-order partial differential equations. First, we establish the existence of a unique non-negative solution by using the Banach fixed point theorem. Then, we provide optimality conditions by means of normal cone and adjoint system. Finally, we derive the existence of an optimal strategy by means of Ekeland’s variational principle. Here the objective functional represents the net economic benefit yielded from harvesting.

Description: In this paper, we establish a large deviation principle for stochastic models of incompressible second grade fluids. The weak convergence method introduced by Budhiraja and Dupuis (Probab Math Statist 20:39–61, 2000 ) plays an important role.

Description: In this paper we study continuous-time stochastic control problems with both monotone and classical controls motivated by the so-called public good contribution problem. That is the problem of n economic agents aiming to maximize their expected utility allocating initial wealth over a given time period between private consumption and irreversible contributions to increase the level of some public good. We investigate the corresponding social planner problem and the case of strategic interaction between the agents, i.e. the public good contribution game. We show existence and uniqueness of the social planner’s optimal policy, we characterize it by necessary and sufficient stochastic Kuhn–Tucker conditions and we provide its expression in terms of the unique optional solution of a stochastic backward equation. Similar stochastic first order conditions prove to be very useful for studying any Nash equilibria of the public good contribution game. In the symmetric case they allow us to prove (qualitative) uniqueness of the Nash equilibrium, which we again construct as the unique optional solution of a stochastic backward equation. We finally also provide a detailed analysis of the so-called free rider effect.

Description: We study optimal control problems in infinite horizon whxen the dynamics belong to a specific class of piecewise deterministic Markov processes constrained to star-shaped networks (corresponding to a toy traffic model). We adapt the results in Soner (SIAM J Control Optim 24(6):1110–1122, 1986 ) to prove the regularity of the value function and the dynamic programming principle. Extending the networks and Krylov’s “shaking the coefficients” method, we prove that the value function can be seen as the solution to a linearized optimization problem set on a convenient set of probability measures. The approach relies entirely on viscosity arguments. As a by-product, the dual formulation guarantees that the value function is the pointwise supremum over regular subsolutions of the associated Hamilton–Jacobi integrodifferential system. This ensures that the value function satisfies Perron’s preconization for the (unique) candidate to viscosity solution.

Description: This paper deals with a class of optimal control problems governed by an initial-boundary value problem of a parabolic equation. The case of semi-linear boundary control is studied where the control is applied to the system via the Wentzell boundary condition. The differentiability of the state variable with respect to the control is established and hence a necessary condition is derived for the optimal solution in the case of both unconstrained and constrained problems. The condition is also sufficient for the unconstrained convex problems. A second order condition is also derived.

Description: 〈h3〉Abstract〈/h3〉 〈p〉In a Hilbert space 〈span〉 〈span〉\({{\mathcal {H}}}\)〈/span〉 〈/span〉, we study the convergence properties of a class of relaxed inertial forward–backward algorithms. They aim to solve structured monotone inclusions of the form 〈span〉 〈span〉\(Ax + Bx \ni 0\)〈/span〉 〈/span〉 where 〈span〉 〈span〉\(A:{{\mathcal {H}}}\rightarrow 2^{{\mathcal {H}}}\)〈/span〉 〈/span〉 is a maximally monotone operator and 〈span〉 〈span〉\(B:{{\mathcal {H}}}\rightarrow {{\mathcal {H}}}\)〈/span〉 〈/span〉 is a cocoercive operator. We extend to this class of problems the acceleration techniques initially introduced by Nesterov, then developed by Beck and Teboulle in the case of structured convex minimization (FISTA). As an important element of our approach, we develop an inertial and parametric version of the Krasnoselskii–Mann theorem, where joint adjustment of the inertia and relaxation parameters plays a central role. This study comes as a natural extension of the techniques introduced by the authors for the study of relaxed inertial proximal algorithms. An illustration is given to the inertial Nash equilibration of a game combining non-cooperative and cooperative aspects.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The value function associated with an optimal control problem subject to the Navier–Stokes equations in dimension two is analyzed. Its smoothness is established around a steady state, moreover, its derivatives are shown to satisfy a Riccati equation at the order two and generalized Lyapunov equations at the higher orders. An approximation of the optimal feedback law is then derived from the Taylor expansion of the value function. A convergence rate for the resulting controls and closed-loop systems is demonstrated. 〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We consider an elliptic variational–hemivariational inequality with constraints in a reflexive Banach space, denoted 〈span〉 〈span〉\(\mathcal{P}\)〈/span〉 〈/span〉, to which we associate a sequence of inequalities 〈span〉 〈span〉\(\{\mathcal{P}_n\}\)〈/span〉 〈/span〉. For each 〈span〉 〈span〉\(n\in \mathbb {N}\)〈/span〉 〈/span〉, 〈span〉 〈span〉\(\mathcal{P}_n\)〈/span〉 〈/span〉 is a variational–hemivariational inequality without constraints, governed by a penalty parameter 〈span〉 〈span〉\(\lambda _n\)〈/span〉 〈/span〉 and an operator 〈span〉 〈span〉\(P_n\)〈/span〉 〈/span〉. Such inequalities are more general than the penalty inequalities usually considered in literature which are constructed by using a fixed penalty operator associated to the set of constraints of 〈span〉 〈span〉\(\mathcal{P}\)〈/span〉 〈/span〉. We provide the unique solvability of inequality 〈span〉 〈span〉\(\mathcal{P}_n\)〈/span〉 〈/span〉. Then, under appropriate conditions on operators 〈span〉 〈span〉\(P_n\)〈/span〉 〈/span〉, we state and prove the convergence of the solution of 〈span〉 〈span〉\(\mathcal{P}_n\)〈/span〉 〈/span〉 to the solution of 〈span〉 〈span〉\(\mathcal{P}\)〈/span〉 〈/span〉. This convergence result extends the results previously obtained in the literature. Its generality allows us to apply it in various situations which we present as examples and particular cases. Finally, we consider a variational–hemivariational inequality with unilateral constraints which arises in Contact Mechanics. We illustrate the applicability of our abstract convergence result in the study of this inequality and provide the corresponding mechanical interpretations.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Trailing stop is a popular stop-loss trading strategy by which the investor will sell the asset once its price experiences a pre-specified percentage drawdown. In this paper, we study the problem of timing to buy and then sell an asset subject to a trailing stop. Under a general linear diffusion framework, we study an optimal double stopping problem with a random path-dependent maturity. Specifically, we first analytically solve the optimal liquidation problem with a trailing stop, and in turn derive the optimal timing to buy the asset. Our method of solution reduces the problem of determining the optimal trading regions to solving the associated differential equations. For illustration, we implement an example and conduct a sensitivity analysis under the exponential Ornstein–Uhlenbeck model.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉This paper presents a widely applicable approach to solving (multi-marginal, martingale) optimal transport and related problems via neural networks. The core idea is to penalize the optimization problem in its dual formulation and reduce it to a finite dimensional one which corresponds to optimizing a neural network with smooth objective function. We present numerical examples from optimal transport, martingale optimal transport, portfolio optimization under uncertainty and generative adversarial networks that showcase the generality and effectiveness of the approach.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Within the framework of the cumulative prospective theory of Kahneman and Tversky, this paper considers a continuous-time behavioral portfolio selection problem whose model includes both running and terminal terms in the objective functional. Despite the existence of 〈em〉S〈/em〉-shaped utility functions and probability distortions, a necessary condition for the optimality is derived. The results are applied to a few examples.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Within the framework of the cumulative prospective theory of Kahneman and Tversky, this paper considers a continuous-time behavioral portfolio selection problem whose model includes both running and terminal terms in the objective functional. Despite the existence of 〈em〉S〈/em〉-shaped utility functions and probability distortions, a necessary condition for the optimality is derived. The results are applied to a few examples.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We study the extension of the proximal gradient algorithm where only a stochastic gradient estimate is available and a relaxation step is allowed. We establish convergence rates for function values in the convex case, as well as almost sure convergence and convergence rates for the iterates under further convexity assumptions. Our analysis avoid averaging the iterates and error summability assumptions which might not be satisfied in applications, e.g. in machine learning. Our proofing technique extends classical ideas from the analysis of deterministic proximal gradient algorithms.〈/p〉

Description: We study pseudomonotone and quasimonotone quasivariational inequalities in a finite dimensional space. In particular we focus our attention on the closedness of some solution maps associated to a parametric quasivariational inequality. From this study we derive two results on the existence of solutions of the quasivariational inequality. On the one hand, assuming the pseudomonotonicity of the operator, we get the nonemptiness of the set of the classical solutions. On the other hand, we show that the quasimonoticity of the operator implies the nonemptiness of the set of nonzero solutions. An application to traffic network is also considered.

Description: We consider a backward stochastic differential equation with jumps (BSDEJ) which is driven by a Brownian motion and a Poisson random measure. We present two candidate-approximations to this BSDEJ and we prove that the solution of each candidate-approximation converges to the solution of the original BSDEJ in a space which we specify. We use this result to investigate in further detail the consequences of the choice of the model to (partial) hedging in incomplete markets in finance. As an application, we consider models in which the small variations in the price dynamics are modeled with a Poisson random measure with infinite activity and models in which these small variations are modeled with a Brownian motion or are cut off. Using the convergence results on BSDEJs, we show that quadratic hedging strategies are robust towards the approximation of the market prices and we derive an estimation of the model risk.

Description: This paper stems from the idea of adopting a new approach to solve some classical optimal packing problems for balls. In fact, we attack this kind of problems (which are of discrete nature) by means of shape optimization techniques, applied to suitable \(\Gamma \) -converging sequences of energies associated to Cheeger type problems. More precisely, in a first step we prove that different optimal packing problems are limits of sequences of optimal clusters associated to the minimization of energies involving suitable (generalized) Cheeger constants. In a second step, we propose an efficient phase field approach based on a multiphase \(\Gamma \) -convergence result of Modica–Mortola type, in order to compute those generalized Cheeger constants, their optimal clusters and, as a consequence of the asymptotic result, optimal packings. Numerical experiments are carried over in two and three space dimensions. Our continuous shape optimization approach to solve discrete packing problems circumvents the NP-hard character of these ones, and efficiently leads to configurations close to the global minima.

Description: We consider the inverse multiphase Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundaries. Optimal control framework is pursued, where boundary heat flux is the control, and the optimality criteria consist of the minimization of the \(L_2\) -norm declination of the trace of the solution to the Stefan problem from the temperature measurement on the fixed right boundary. The state vector solves multiphase Stefan problem in a weak formulation, which is equivalent to Neumann problem for the quasilinear parabolic PDE with discontinuous coefficient. Full discretization through finite differences is implemented and discrete optimal control problem is introduced. We prove well-posedness in a Sobolev space framework and convergence of discrete optimal control problems to the original problem both with respect to the cost functional and control. Along the way, the convergence of the method of finite differences for the weak solution of the multiphase Stefan problem is proved. The proof is based on achieving a uniform \(L_{\infty }\) bound, and \(W_2^{1,1}\) -energy estimate for the discrete multiphase Stefan problem.

Description: A parametric constrained convex optimal control problem, where the initial state is perturbed and the linear state equation contains a noise, is considered in this paper. Formulas for computing the subdifferential and the singular subdifferential of the optimal value function at a given parameter are obtained by means of some recent results on differential stability in mathematical programming. The computation procedures and illustrative examples are presented.

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this paper, we study the split common null point problem. Then, using the hybrid projection method and the metric resolvent of monotone operators, we prove a strong convergence theorem for an iterative method for finding a solution of this problem in Banach spaces.〈/p〉

Description: 〈p〉The original version of this article unfortunately contained a few mistakes in Theorems and notation. The corrected information is given below.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉By a memory mean-field process we mean the solution 〈span〉 〈span〉\(X(\cdot )\)〈/span〉 〈/span〉 of a stochastic mean-field equation involving not just the current state 〈em〉X〈/em〉(〈em〉t〈/em〉) and its law 〈span〉 〈span〉\(\mathcal {L}(X(t))\)〈/span〉 〈/span〉 at time 〈em〉t〈/em〉,  but also the state values 〈em〉X〈/em〉(〈em〉s〈/em〉) and its law 〈span〉 〈span〉\(\mathcal {L}(X(s))\)〈/span〉 〈/span〉 at some previous times 〈span〉 〈span〉\(s〈t.\)〈/span〉 〈/span〉 Our purpose is to study stochastic control problems of memory mean-field processes. We consider the space 〈span〉 〈span〉\(\mathcal {M}\)〈/span〉 〈/span〉 of measures on 〈span〉 〈span〉\(\mathbb {R}\)〈/span〉 〈/span〉 with the norm 〈span〉 〈span〉\(|| \cdot ||_{\mathcal {M}}\)〈/span〉 〈/span〉 introduced by Agram and Øksendal (Model uncertainty stochastic mean-field control. 〈a href="http://arxiv.org/abs/1611.01385v5"〉arXiv:1611.01385v5〈/a〉, [〈span〉2〈/span〉]), and prove the existence and uniqueness of solutions of memory mean-field stochastic functional differential equations. We prove two stochastic maximum principles, one sufficient (a verification theorem) and one necessary, both under partial information. The corresponding equations for the adjoint variables are a pair of (〈em〉time〈/em〉-〈em〉advanced backward stochastic differential equations〈/em〉 (absdes), one of them with values in the space of bounded linear functionals on path segment spaces. As an application of our methods, we solve a memory mean–variance problem as well as a linear–quadratic problem of a memory process.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The theory of micropolar fluids emphasizes the micro-structure of fluids by coupling the Navier–Stokes equations with micro-rotational velocity, and is widely viewed to be well fit, better than the Navier–Stokes equations, to describe fluids consisting of bar-like elements such as liquid crystals made up of dumbbell molecules or animal blood. Following the work of Weinan et al. (Commun Math Phys 224:83–106, 〈span〉2001〈/span〉), we prove the existence of a unique stationary measure for the stochastic micropolar fluid system with periodic boundary condition, forced by only the determining modes of the noise and therefore a type of finite-dimensionality of micropolar fluid flow. The novelty of the manuscript is a series of energy estimates that is reminiscent from analysis in the deterministic case.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We consider the initial boundary value problem for a class of logarithmic wave equations with linear damping. By constructing a potential well and using the logarithmic Sobolev inequality, we prove that, if the solution lies in the unstable set or the initial energy is negative, the solution will grow as an exponential function in the 〈span〉 〈span〉\(H^1_0(\Omega )\)〈/span〉 〈/span〉 norm as time goes to infinity. If the solution lies in a smaller set compared with the stable set, we can estimate the decay rate of the energy. These results are extensions of earlier results.〈/p〉

Description: 〈p〉The original version of this article unfortunately contained a mistake in the equation.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this paper we study the standard optimal control and time optimal control problems for a class of semilinear evolution systems with infinite delay. We first establish the results of existence and uniqueness of mild solution and the compactness of the solution operator for the control system. Then, based on these results, we investigate the optimal control problem with integral cost function and the time optimal control problem respectively. Under some conditions we show the existence of optimal controls for the both cases of bounded and unbounded admissible control sets. We also obtain the existence of time optimal control to a target set. In addition, a convergence theorem of time optimal controls to a point target set is proved. Finally, an example is given to show the application of the main results.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We investigate the long-time dynamics and optimal control problem of a thermodynamically consistent diffuse interface model that describes the growth of a tumor in presence of a nutrient and surrounded by host tissues. The state system consists of a Cahn–Hilliard type equation for the tumor cell fraction and a reaction–diffusion equation for the nutrient. The possible medication that serves to eliminate tumor cells is in terms of drugs and is introduced into the system through the nutrient. In this setting, the control variable acts as an external source in the nutrient equation. First, we consider the problem of “long-time treatment” under a suitable given mass source and prove the convergence of any global solution to a single equilibrium as 〈span〉 〈span〉\(t\rightarrow +\infty \)〈/span〉 〈/span〉. Second, we consider the “finite-time treatment” that corresponds to an optimal control problem. Here we allow the objective cost functional to depend on a free time variable, which represents the unknown treatment time to be optimized. We prove the existence of an optimal control and obtain first order necessary optimality conditions for both the drug concentration and the treatment time. One of the main aim of the control problem is to realize in the best possible way a desired final distribution of the tumor cells, which is expressed by the target function 〈span〉 〈span〉\(\phi _\Omega \)〈/span〉 〈/span〉. By establishing the Lyapunov stability of certain equilibria of the state system (without external source), we show that 〈span〉 〈span〉\(\phi _{\Omega }\)〈/span〉 〈/span〉 can be taken as a stable configuration, so that the tumor will not grow again once the finite-time treatment is completed.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Recent results have proven the minimax optimality of LASSO and related algorithms for noisy linear regression. However, these results tend to rely on variance estimators that are inefficient or optimizations that are slower than LASSO itself. We propose an efficient estimator for the noise variance in high dimensional linear regression that is faster than LASSO, only requiring 〈em〉p〈/em〉 matrix–vector multiplications. We prove this estimator is consistent with a good rate of convergence, under the condition that the design matrix satisfies the restricted isometry property (RIP). In practice, our estimator scales incredibly well into high dimensions, is highly parallelizable, and only incurs a modest bias.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In the present paper, we focus on the vector optimization problems with constraints, where objective functions and constrained functions are Fréchet differentiable, and whose gradient mapping is locally Lipschitz. By using the second-order symmetric subdifferential and the second-order tangent set, we introduce some new types of second-order regularity conditions in the sense of Abadie. Then we establish some second-order necessary optimality conditions Karush–Kuhn–Tucker-type for local efficient (weak efficient, Geoffrion properly efficient) solutions of the considered problem. In addition, we provide some sufficient conditions for a local efficient solution to such problem.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We consider an optimal control problem for the 3D Navier–Stokes–Voigt equations in bounded domains, where the time needed to reach a desired state plays an essential role. We first prove the existence of optimal solutions, and then establish the first-order necessary optimality conditions and the second-order sufficient optimality conditions.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉A stochastic control representation for solution of the Schrödinger equation is obtained, utilizing complex-valued diffusion processes. The Maslov dequantization is employed, where the domain is complex-valued in the space variable. The notion of stationarity is utilized to relate the Hamilton–Jacobi form of the dequantized Schrödinger equation to its stochastic control representation. Convexity is not required, and consequently, there is no restriction on the duration of the problem. Additionally, existence is reduced to a real-valued domain case.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We study a zero-sum stochastic differential game (SDG) in which one controller plays an impulse control while their opponent plays a stochastic control. We consider an asymmetric setting in which the impulse player commits to, at the start of the game, performing less than 〈em〉q〈/em〉 impulses (〈em〉q〈/em〉 can be chosen arbitrarily large). In order to obtain the uniform continuity of the value functions, previous works involving SDGs with impulses assume the cost of an impulse to be decreasing in time. Our work avoids such restrictions by requiring impulses to occur at rational times. We establish that the resulting game admits a value, and in turn, the existence and uniqueness of viscosity solutions to an associated Hamilton-Jacobi-Bellman-Isaacs quasi-variational inequality.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this paper, by introducing a new concept of the (〈em〉f〈/em〉, 〈em〉g〈/em〉, 〈em〉h〈/em〉)-quasimonotonicity and applying the maximal monotonicity of bifunctions and KKM technique, we show the existence results of solutions for quasi mixed equilibrium problems when the constraint set is compact, bounded and unbounded, respectively, which extends and improves several well-known results in many respects. Next, we also obtain a result of optimal control to a minimization problem. Our main results can be applied to the problems of evolution equations, differential inclusions and hemivariational inequalities.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this paper we analyze an homogeneous and isotropic mixture of viscoelastic solids. We propose conditions to guarantee the coercivity of the internal energy and also of the dissipation, first in dimension two and later in dimension three. We obtain an uniqueness result for the solutions when the dissipation is positive and without any hypothesis over the internal energy. When the internal energy and the dissipation are both positive, we prove the existence of solutions as well as their analyticity. Exponential stability and impossibility of localization of the solutions are immediate consequences.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉This paper investigates the optimal contraception control for a nonlinear size-structured population model with three kinds of mortality rates: intrinsic, intra-competition and female sterilant. First, we transform the model to a system of two subsystems, and establish the existence of a unique non-negative solution by means of frozen coefficients and fixed point theory, and show the continuous dependence of the population density on control variable. Then, the existence of an optimal control strategy is proved via compactness and extremal sequence. Next, necessary optimality conditions of first order are established in the form of an Euler–Lagrange system by the use of tangent-normal cone technique and adjoint system. Moreover, a numerical result for the optimal control strategy is presented. Our conclusions would be useful for managing the vermin.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this work we first present the existence, uniqueness and regularity of the strong solution of the tidal dynamics model perturbed by Lévy noise. Monotonicity arguments have been exploited in the proofs. We then formulate a martingale problem of Stroock and Varadhan associated to an initial value control problem and establish existence of optimal controls.〈/p〉