Applied Mathematics and Optimization最新文献

A Neumann-type initial-boundary value problem for

is considered in a smoothly bounded domain (Omega subset mathbb {R}^n), (nge 1). In the case when (n=1), (gamma equiv Gamma ) and (fequiv F), this system coincides with the standard model for heat generation in a viscoelastic material of Kelvin-Voigt type, well-understood in situations in which (gamma =const). Covering scenarios in which all key ingredients (gamma ,Gamma ,f) and F may depend on the temperature (Theta ) here, for initial data which merely satisfy (u_0in W^{1,p+2}(Omega )), (u_{0t}in W^{1,p}(Omega )) and (Theta _0in W^{1,p}(Omega )) with some (pge 2) such that (p>n), a result on local-in-time existence and uniqueness is derived in a natural framework of weak solvability.

This paper investigates the stochastic linear-quadratic (LQ, for short) optimal control problems with random coefficients and regime switching in a finite time horizon where the state equation is multi-dimensional. Similar to the classical stochastic LQ problems, we establish the relationship between the stochastic LQ optimal control problems with regime switching and the related extended stochastic Riccati equations. To solve the extended stochastic Riccati equations, we construct a monotone Piccard iterative sequence and present the link between this sequence and solutions of a family of forward-backward stochastic differential equations. Relying on (L^p) estimates for FBSDEs, we show that the extended stochastic Riccati equation has a solution. This partially addresses one question left in Hu et al. (Ann. Appl. Probab. 32(1): 426-460, 2022). Finally, the stochastic LQ optimal control problems with regime switching is solved.

We study the null controllability for a degenerate/singular wave equation with drift in non divergence form. In particular, considering a control localized on the non degenerate boundary point, we provide some conditions for the boundary controllability via energy methods and boundary observability.

A generalized version of an important theorem called Hoffman’s lemma in the book by Bonnans and Shapiro (Perturbation analysis of optimization problems, Springer, Berlin, 2000), which deals with generalized polyhedral convex multifunctions, is obtained in this paper. Under a mild assumption, the result allows us to demonstrate that the domain of a generalized polyhedral convex multifunction is closed and the multifunction is Lipschitz continuous on its effective domain. As concrete applications of the results, we prove some local error bounds for generalized affine variational inequalities and a theorem on the (strong) convergence of feasible descent methods for solving generalized quadratic programming problems.

We investigate the systematic design of compliant morphing structures composed of materials reacting to an external stimulus. We add a perimeter penalty term to ensure existence of solutions. We propose a phase-field approximation of this sharp interface problem, prove its convergence as the regularization length approaches 0 and present an efficient numerical implementation. We illustrate the strengths of our approach through a series of numerical examples.

We consider a Cahn–Hilliard–Darcy system for an incompressible mixture of two fluids we already analyzed in [9]. In this system, the relative concentration difference (varphi ) obeys a convective nonlocal Cahn–Hilliard equation with degenerate mobility and singular (e.g., logarithmic) potential, while the volume averaged fluid velocity (varvec{u}) is given by a Darcy’s law subject to the Korteweg force (mu nabla varphi ), where the chemical potential (mu ) is defined by means of a nonlocal Helmholtz free energy. The kinematic viscosity (eta ) depends on (varphi ). With respect to the quoted contribution, here we assume that the Darcy’s law is subject to gravity and to a given additional source. Moreover, we suppose that the Cahn–Hilliard equation and the chemical potential contain source terms. Our main goal is to establish the existence of two notions of weak solutions. The first, called “generalized” weak solution, is based a convenient splitting of (mu ) so that the entropy derivative does not need to be integrable. The second is slightly stronger and allows to reconstruct (mu ) and to prove the validity of a canonical energy identity. For this reason, the latter is called “natural” weak solution. The rigorous relation between the two notions of weak solution is also analyzed. The existence of a global attractor for generalized weak solutions and time independent sources is then demonstrated via the theory of generalized semiflows introduced by J.M. Ball.

In this paper, we study a general mean-field partial information non-zero sum stochastic differential game, in which the dynamic of state is described by a stochastic differential equation (SDE) depending on the distribution of the state and the control domain of each player can be non-convex. Moreover, the control variables of both players can enter the diffusion coefficients of the state equation. We establish a necessary condition in the form of Pontryagin’s maximum principle for optimality. Then a verification theorem is obtained for optimal control when the control domain is convex. As an application, our results are applied to studying linear–quadratic (LQ) mean-field game in the type of scalar interaction.

In this work, we consider a nonsmooth minimisation problem in which the objective function can be represented as the maximum of finitely many smooth “component functions”. First, we study a smooth min–max reformulation of the problem. Due to this smoothness, the model provides enhanced capability of exploiting the structure of the problem, when compared to methods that attempt to tackle the nonsmooth problem directly. Then, we present several approaches to identify the set of active component functions at a minimiser, all within finitely many iterations of a first order method for solving the smooth model. As is well known, the problem can be equivalently rewritten in terms of these component functions, but a key challenge is to identify this set a priori. Such an identification is clearly beneficial in an algorithmic sense, since we can discard those component functions which are not necessary to describe the solution, which in turn can facilitate faster convergence. Finally, numerical results comparing the accuracy of each of these approaches are presented, along with the effect they have on reducing the complexity of the original problem.

We establish existence and uniqueness of minimax solutions for a fairly general class of path-dependent Hamilton–Jacobi equations. In particular, the relevant Hamiltonians can contain the solution and they only need to be measurable with respect to time. We apply our results to optimal control problems of (delay) functional differential equations with cost functionals that have discount factors and with time-measurable data. Our main results are also crucial for our companion paper Bandini and Keller (Non-local Hamilton–Jacobi–Bellman equations for the stochastic optimal control of path-dependent piecewise deterministic processes, 2024, http://arxiv.org/abs/2408.02147), where non-local path-dependent Hamilton–Jacobi–Bellman equations associated to the stochastic optimal control of non-Markovian piecewise deterministic processes are studied.