Applied Mathematics and Optimization最新文献

The focus of this paper is on the null controllability of two kinds of coupled systems including both degenerate and non-degenerate equations with switching control. We first establish the observability inequality for measurable subsets in time for such coupled system, and then by the HUM method to obtain the null controllability. Next, we investigate the null controllability of such coupled system for segmented time intervals. Notably, these results are obtained through spectral inequalities rather than using the method of Carleman estimates. Such coupled systems with switching control, to the best of our knowledge, are among the first to discuss.

This paper is concerned with the pullback measure attractors of the non-autonomous fractional reaction-diffusion equations defined on (mathbb {R}^{n}). We first prove the existence and uniqueness of pullback measure attractors for such equations. Then we establish the upper semi-continuity of these attractors as the noise intensity (varepsilon ) tends to zero. Specifically, we apply the uniform estimates on the tails of solutions to prove the asymptotic compactness of a family of probability distributions of solutions to overcome the non-compactness of usual Sobolev embeddings on unbounded domains.

The paper investigates the global well-posedness and the longtime dynamics for a class of strongly damped wave equations with evolutional p(x, t)-Laplacian and q(x, t)-growth source term on a bounded domain ( Omega subset {mathbb {R}}^3: u_{tt}-nabla cdot (|nabla u|^{p(x, t)-2} nabla u)-lambda Delta u- Delta u_t+ |u|^{q(x, t)-2}u=g), together with the perturbed parameter (lambda in [0,1]) and the Dirichlet boundary condition. We show that under rather relaxed conditions, (i) the model is global well-posed; (ii) for each (lambda _0in (0,1]), the related nonautonomous dynamical systems acting on the time-dependent phase spaces have a family of pullback ({mathscr {D}})-exponential attractor ({mathcal {E}}_lambda ={E_lambda (t)}_{tin {mathbb {R}}}in {mathscr {D}}) which is Hölder continuous w.r.t. (lambda ) at (lambda _0); (iii) they have also a family of finite dimensional pullback ({mathscr {D}})-attractors ({mathcal {A}}_lambda ={A_lambda (t)}_{tin {mathbb {R}}}) which are upper semicontinuous and residual continuous w.r.t. (lambda in (0,1]). In particular, when (lambda in (0,1]) and without the p(x, t)-Laplacian, the above mentioned results can be greatly improved, in the concrete; (iv) the weak solutions of the corresponding model possess additionally partial regularity and the Hölder stability in stronger (H^1times H^1)-norm, the pullback ({mathscr {D}})-attractor and pullback ({mathscr {D}})-exponential attractor in weaker ({mathcal {Y}}_1)-norm can be regularized to be those in stronger (H^1times H^1)-norm, which are also the standard ones in ({mathcal {H}}_t)-norm. The method provided here allows overcoming the difficulties arising from variable exponent nonlinearities and extending the analysis and the results for these type of models with constant exponent nonlinearities.

We address a system of equations modeling an incompressible fluid interacting with an elastic body. We prove the local existence when the initial velocity belongs to the space (H^{1.5+epsilon }) and the initial structure velocity is in (H^{1+epsilon }), where (epsilon in (0, 1/20)).

We introduce a non-zero-sum game between a government and a legislative body to study the optimal level of debt. Each player, with different time preferences, can intervene on the stochastic dynamics of the debt-to-GDP ratio via singular stochastic controls, in view of minimizing non-continuously differentiable running costs. We completely characterise Nash equilibria in the class of Skorokhod-reflection-type policies. We highlight the importance of different time preferences resulting in qualitatively different type of equilibria. In particular, we show that, while it is always optimal for the government to devise an appropriate debt issuance policy, the legislator should optimally impose a debt ceiling only under relatively low discount rates and a laissez-faire policy can be optimal for high values of the legislator’s discount rate.

This paper extends the results provided in Jasso-Fuentes et al. (Appl Math Optim 81(2):409–441, 2020b) and Jasso-Fuentes et al. (Pure Appl Funct Anal 9(3):675–704, 2024) regarding the study of discrete-time hybrid stochastic models with general spaces and total discounted payoffs. This extension incorporates the handling of negative and/or unbounded costs per stage. In particular, it encompasses interesting applications, such as scenarios where the controller optimizes net costs, social welfare costs, or distances between points. These situations arise when assumptions of both non-negativeness and boundedness on the cost per stage do not apply. Our proposal relies on Lyapunov-like conditions, enabling, among other aspects, the finiteness of the value function and the existence of solutions to the associated dynamic programming equation. This equation is crucial for deriving optimal control policies. To illustrate our theory, we include an example in inventory-manufacturing management, highlighting its evident hybrid nature.

To study the nonlinear properties of complex natural phenomena, the evolution of the quantity of interest can be often represented by systems of coupled nonlinear stochastic differential equations (SDEs). These SDEs typically contain several parameters which have to be chosen carefully to match the experimental data and to validate the effectiveness of the model. In the present paper the calibration of these parameters is described by nonlinear SDE-constrained optimization problems. In the optimize-before-discretize setting a rigorous analysis is carried out to ensure the existence of optimal solutions and to derive necessary first-order optimality conditions. For the numerical solution a Monte–Carlo method is applied using parallelization strategies to compensate for the high computational time. In the numerical examples an Ornstein–Uhlenbeck and a stochastic Prandtl–Tomlinson bath model are considered.

A class of parametric optimal control problems governed by semilinear parabolic equations with mixed pointwise constraints is investigated. The perturbations appear in the objective functional, the state equation and in mixed pointwise constraints. By analyzing regularity and establishing stability condition of Lagrange multipliers we prove that, if the unperturbed problem satisfies the strong second-order sufficient condition, then the solution map and the associated Lagrange multipliers are locally Lipschitz continuous functions of parameters.

The primary objective of this paper is to study a new class of parabolic quasi variational–hemivariational inequalities. First, we prove a unique solvability result for such class under some mild conditions. Second, we show the existence of an optimal solution for an associated control problem. Finally, these results are applied to a model of quasistatic frictional contact in mechanics.

In this paper we study the optimal control of a parabolic initial-boundary value problem of viscous Cahn–Hilliard type with zero Neumann boundary conditions. Phase field systems of this type govern the evolution of diffusive phase transition processes with conserved order parameter. It is assumed that the nonlinear functions driving the physical processes within the spatial domain are double-well potentials of logarithmic type whose derivatives become singular at the boundary of their respective domains of definition. For such systems, optimal control problems have been studied in the past. We focus here on the situation when the cost functional of the optimal control problem contains a nondifferentiable term like the (L^1)-norm, which leads to sparsity of optimal controls. For such cases, we establish first-order necessary and second-order sufficient optimality conditions for locally optimal controls. In the approach to second-order sufficient conditions, the main novelty of this paper, we adapt a technique introduced by Casas et al. in the paper (SIAM J Control Optim 53:2168–2202, 2015). In this paper, we show that this method can also be successfully applied to systems of viscous Cahn–Hilliard type with logarithmic nonlinearity. Since the Cahn–Hilliard system corresponds to a fourth-order partial differential equation in contrast to the second-order systems investigated before, additional technical difficulties have to be overcome.