Applied Mathematics and Optimization最新文献

We investigate the convergence, in the small mass limit, of the stationary solutions of a class of stochastic damped wave equations, where the friction coefficient depends on the state and the noisy perturbation is of multiplicative type. We show that the Smoluchowski–Kramers approximation that has been previously shown to be true in any fixed time interval, is still valid in the long time regime. Namely, we prove that the first marginals of any sequence of stationary solutions for the damped wave equation converge to the unique invariant measure of the limiting stochastic quasilinear parabolic equation. The convergence is proved with respect to the Wasserstein distance associated with the (H^{-1}) norm.

In this paper we investigate optimal control problems perturbed by random events. We assume that the control has to be decided prior to observing the outcome of the perturbed state equations. We investigate the use of probability functions in the objective function or constraints to define optimal or feasible controls. We provide an extension of differentiability results for probability functions in infinite dimensions usable in this context. These results are subsequently combined with the optimal control setting to derive a novel Pontryagin’s optimality principle.

In this work, we focus on an infinite horizon mean-field linear-quadratic stochastic control problem with jumps. Firstly, the infinite horizon linear mean-field stochastic differential equations and backward stochastic differential equations with jumps are studied to support the research of the control problem. The global integrability properties of their solution processes are studied by introducing a kind of so-called dissipation conditions suitable for the systems involving the mean-field terms and jumps. For the control problem, we conclude a sufficient and necessary condition of open-loop optimal control by the variational approach. Besides, a kind of infinite horizon fully coupled linear mean-field forward-backward stochastic differential equations with jumps is studied by using the method of continuation. Such a research makes the characterization of the open-loop optimal controls more straightforward and complete.

Motivated by the self-pursuit of controlled objects, we consider the exact controllability of a linear mean-field type game-based control system (MF-GBCS, for short) generated by a linear-quadratic (LQ, for short) Nash game. A Gram-type criterion for the general time-varying coefficients case and a Kalman-type criterion for the special time-invariant coefficients case are obtained. At the same time, the equivalence between the exact controllability of this MF-GBCS and the exact observability of a dual system is established. Moreover, an admissible control that can steer the state from any initial vector to any terminal random variable is constructed in closed form.

Dynamic programming equations for mean field control problems with a separable structure are Eikonal type equations on the Wasserstein space. Standard differentiation using linear derivatives yield a direct extension of the classical viscosity theory. We use Fourier representation of the Sobolev norms on the space of measures, together with the standard techniques from the finite dimensional theory to prove a comparison result among semi-continuous sub and super solutions, obtaining a unique characterization of the value function.

We are concerned with a third order in time equation in the presence of viscoelastic effects given by the memory term and with a semi linear source term, posed on a bounded domain (Omega subset mathbb {R}^3 ). Considering three different types of memory in the past history framework, we prove the well-posedness of its solutions as well as the exponential stability of the energy functional. Relaxing some hypotheses on the memory kernel, we improve and extend the results established in the existing literature.

We study the local and global existence and uniqueness of a strict solution for a general class of abstract explicit neutral equations with state-dependent delay. Some examples on explicit partial neutral differential equations with state dependent delay are presented.

The study of the paper mainly focuses on recovering the dissipative parameter in a coupled system formed by coupling a bilaplacian operator to a heat equation from final time measured output data via a quasi-solution approach with optimization. The inverse coefficient problem is expressed as a minimization problem. We establish the existence of a minimizer and extract the necessary optimality condition, which is essential in proving the requisite stability result for the inverse coefficient problem. The effectiveness of the proposed approach is demonstrated through an analysis of numerical results using the conjugate gradient approach.

This paper is devoted to the existence and uniqueness of solution for a large class of perturbed sweeping processes formulated by fractional differential inclusions in infinite dimensional setting. The normal cone to the (mildly non-convex) prox-regular moving set C(t) is supposed to have a Hölder continuous variation, is perturbed by a continuous mapping, which is both time and state dependent. Using an explicit catching-up algorithm, we show that the fractional perturbed sweeping process has one and only one Hölder continuous solution. Then this abstract result is applied to provide a theorem on the weak solvability of a fractional viscoelastic frictionless contact problem. The process is quasistatic and the constitutive relation is modeled with the fractional Kelvin–Voigt law. This application represents an additional novelty of our paper.

In this paper we study an optimal control problem associated to a parabolic–elliptic chemo-repulsion system with a linear production term in a two-dimensional domain. Under the injection/extract chemical substance on a control subdomain (varOmega _c), we prove the existence and uniqueness of global-in-time strong solutions. Afterwards, for the optimal control problem, we prove the existence of at least one global optimal solution, and derive an optimality system via using a Lagrange multipliers theorem.