Applied Mathematics and Optimization最新文献

This paper is concerned with a three-component chemotaxis model accounting for indirect signal production, reading as (u_t = nabla cdot (nabla u - unabla v)), (v_t = Delta v - v + w) and (0 = Delta w - w + u), posed in a ball of (mathbb {R}^n) with (nge 5), subject to homogeneous Neumann boundary conditions. The system is a Nagai-type variant of its fully parabolic version that has a four-dimensional critical mass phenomenon concerning blowup in finite or infinite time according to the seminal works of Fujie and Senba [J. Differential Equations, 263 (2017), 88–148; 266 (2019), 942–976]. We prove that for any prescribed mass (m > 0), there exist radially symmetric and positive initial data ((u_0,v_0)in C^0(overline{Omega })times C^2(overline{Omega })) with (int _Omega u_0 = m) such that the corresponding solutions blow up in finite time.

We consider an optimal control problem where the state is governed by a free boundary problem called the two-phase membrane problem and the control appears in the coefficients of the state equation, influencing the positive and negative phases of the solution. Our investigation focuses on various properties associated with the control-to-state map. Due to the non-differentiability of this map, we regularize the state equation. The existence, uniqueness, and characterization of the optimal pairs are established.

The aim of this paper is to study a stochastic unbounded delay evolution variational inequality which consists of a stochastic unbounded delay evolution equation driven by tempered fractional noise with exponential dichotomy and a stochastic variational inequality. First, the existence and uniqueness of the mild solution on ( {mathbb {R}} ) are established for the linear stochastic evolution equation overcoming the challenges posed by the exponential dichotomy of the evolution family ( left{ S(t,s)right} _{tge s} ) generated by the family of closed, densely defined linear operators A(t) in (1). Then after giving the equivalent form of the stochastic variational inequality defined on ( {mathbb {R}} ), the existence and uniqueness of the mild solution on ( {mathbb {R}} ) are proved for the stochastic unbounded delay evolution variational inequality (1) by using the Banach fixed point theorem instead of the iteration method and convergence analysis. Notably, due to the nontrivial exponential dichotomy of the evolution family ( left{ S(t,s)right} _{tge s} ), the stability can not be established for the nonlinear stochastic evolution variational inequality (1) and even for the linear stochastic evolution equation (5). Moreover, we show the exponential stability of the nontrivial equilibrium solution for the stochastic unbounded delay evolution variational inequality (1) but under the assumption that the evolution family ( left{ S(t,s)right} _{tge s} ) is exponential stable. Finally, the stochastic reaction diffusion variational inequality is considered as an example of application.

We study a multi-agent mean-field type control problem in discrete time where the agents aim to find a socially optimal strategy and where the state and action spaces for the agents are assumed to be continuous. The agents are only weakly coupled through the distribution of their state variables. The problem in its original form can be formulated as a classical Markov decision process (MDP), however, this formulation suffers from several practical difficulties. In this work, we attempt to overcome the curse of dimensionality, coordination complexity between the agents, and the necessity of perfect feedback collection from all the agents (which might be hard to do for large populations.) We provide several approximations: we establish the near optimality of the action and state space discretization of the agents under standard regularity assumptions for the considered formulation by constructing and studying the measure valued MDP counterpart for finite and infinite population settings. It is a well known approach to consider the infinite population problem for mean-field type models, since it provides symmetric policies for the agents which simplifies the coordination between the agents. However, the optimality analysis is harder as the state space of the measure valued infinite population MDP is continuous (even after space discretization of the agents). Therefore, as a final step, we provide two further approximations for the infinite population problem: the first one directly aggregates the probability measure space, and requires the distribution of the agents to be collected and mapped with a nearest neighbor map, and the second method approximates the measure valued MDP through the empirical distributions of a smaller sized sub-population, for which one only needs keep track of the mean-field term as an estimate by collecting the state information of a small sub-population. For each of the approximation methods, we provide provable regret bounds.

In this paper, we investigate a class of variable exponent double phase elliptic inclusion systems involving anisotropic partial differential operators with logarithmic perturbation as well as two fully coupled multivalued terms, one of them is defined in the domain and the other is defined on the boundary, respectively. Firstly, under the suitable coercive conditions, the existence of a weak solution for the double phase elliptic inclusion systems is verified via applying a surjectivity theorem concerning multivalued pseudomonotone operators. Then, when the elliptic inclusion system is considered in non-coercive framework, we employ the sub-supersolution method to establish the existence and compactness results. Finally, we deliver several solvability properties of some special cases with respect to the elliptic inclusion system under consideration via constructing proper sub- and super-solutions.

This paper addresses a new class of generalized Bolza problems governed by nonconvex integro-differential inclusions with endpoint constraints on trajectories, where the integral terms are given in the general (with time-dependent integrands in the dynamics) Volterra form. We pursue here a threefold goal. First we construct well-posed approximations of continuous-time integro-differential systems by their discrete-time counterparts with showing that any feasible solution to the original system can be strongly approximated in the (W^{1,2})-norm topology by piecewise-linear extensions of feasible discrete trajectories. This allows us to verify in turn the strong convergence of discrete optimal solutions to a prescribed local minimizer for the original problem. Facing intrinsic nonsmoothness of original integro-differential problem and its discrete approximations, we employ appropriate tools of generalized differentiation in variational analysis to derive necessary optimality conditions for discrete-time problems (which is our second goal) and finally accomplish our third goal to obtain necessary conditions for the original continuous-time problems by passing to the limit from discrete approximations. In this way we establish, in particular, a novel necessary optimality condition of the Volterra type, which is the crucial result for dynamic optimization of integro-differential inclusions.

The main purpose of this paper is to give an upper bound of Hausdorff dimension of random attractors for a stochastic delayed parabolic equation in Banach spaces. The estimation of dimensions of random attractors are obtained by combining the squeezing property and a covering lemma of finite subspace of Banach spaces, which generalizes the method established in Hilbert spaces. Due to the lack of smooth inner product geometry structure, we adopt the state decomposition of phase space based on the exponential dichotomy of the linear deterministic part of the studied equations instead of orthogonal projectors with finite ranks used for stochastic partial differential equations. The obtained dimension of the random attractors depends only on the inner characteristics of the studied equation, such as spectrum of the linear part and the random Lipschitz constant of the nonlinear term, while not relating to the compact embedding of the phase space to another Banach space as the existing works did.

The convergence rate in Wasserstein distance is estimated for empirical measures of ergodic Markov processes, and the estimate can be sharp in some specific situations. The main result is applied to subordinations of typical models excluded by existing results, which include: stochastic Hamiltonian systems on ({mathbb{R}}^{n}times {mathbb{R}}^{m}), spherical velocity Langevin processes on ({mathbb{R}}^ntimes mathbb S^{n-1},) multi-dimensional Wright–Fisher type diffusion processes, and stable type jump processes.

We study three types of fourth-order Steklov eigenvalue problems. For the first two of them, we derive the asymptotic expansion of the eigenvalues on Euclidean annular domains (mathbb {B}^n_1setminus overline{mathbb {B}^n_epsilon }) as (epsilon rightarrow 0), in turn yielding some interesting results regarding the shape optimization of the eigenvalues. For these two problems, we also compute the respective spectra on cylinders over closed Riemannian manifolds. For the third problem, we obtain a sharp upper bound for its first non-zero eigenvalue on star-shaped and mean convex Euclidean domains.

In this paper, the study of nonsmooth optimal control problems (P) involving a controlled sweeping process with three main characteristics is launched. First, the sweeping sets are nonsmooth, time-dependent, and uniformly prox-regular. Second, the sweeping process is coupled with a controlled differential equation. Third, a joint-state endpoints constraint set S is present. This general model incorporates different important controlled submodels, such as a class of second order sweeping processes, and coupled evolution variational inequalities. A full form of the nonsmooth Pontryagin maximum principle for strong local minimizers in (P) is derived for bounded or unbounded moving sweeping sets satisfying local constraint qualifications (CQ) without any additional restriction. The existence and uniqueness of a Lipschitz solution for the Cauchy problem of our dynamic is established and the existence of an optimal solution for (P) is obtained. Two of the novelties in achieving the first goal are (i) the construction of a problem over truncated sweeping sets and truncated joint endpoints constraint set that has the same strong local minimizer as (P) and its (CQ) automatically holds, and (ii) the complete redesign of the exponential-penalty approximation technique for problems with moving sweeping sets that do not require any special assumption on the sets, their corners, or on the gradients of their generators. The utility of the optimality conditions is illustrated with an example.