Applied Mathematics and Optimization最新文献

This paper studies the controllability for a Keller–Segel type chemotaxis model with singular sensitivity. Based on the Hopf–Cole transformation, a nonlinear parabolic system, which has first-order couplings, and the coupling coefficients are functions that depend on both time and space variables, is derived. Then, the controllability result is proved by a new global Carleman estimate for general coupled parabolic equations allowed to contain a convective term. Also, the global existence of nonnegative solution for the chemotaxis system is discussed.

In this work, we use the integral definition of the fractional Laplace operator and study a sparse optimal control problem involving a fractional, semilinear, and elliptic partial differential equation as state equation; control constraints are also considered. We establish the existence of optimal solutions and first and second order optimality conditions. We also analyze regularity properties for optimal variables. We propose and analyze two finite element strategies of discretization: a fully discrete scheme, where the control variable is discretized with piecewise constant functions, and a semidiscrete scheme, where the control variable is not discretized. For both discretization schemes, we analyze convergence properties and a priori error bounds.

Based on probabilistic methods, we discuss the relationship between viscosity and distribution solutions for semi-linear partial differential equations (PDEs) with Neumann boundary conditions. We also extend the research to a type of nonlinear PDEs, which is completed through the well-posedness and continuity results of solutions to the corresponding forward-backward SDE.

This paper is concerned with a semilinear Rao–Nakra sandwich beam under the action of three nonlinear localized frictional damping terms in which the core viscoelastic layer is constrained by the pure elasticity or piezoelectric outer layers. The main goal is to prove its asymptotic behavior by applying minimal amount of support to the damping. We firstly prove that the system is global well-posedness by the theory of monotone operators. For asymptotic behavior of solutions, we obtain uniform decay rate results of the system and the energy decay rates are determined by a nonlinear first-order ODE. The existence of a smooth global attractor with finite fractal dimension and generalized exponential attractors are finally obtained.

We derive new necessary and sufficient conditions for strict efficiency in vector optimization problems for non-smooth mappings. Unlike other approaches, our conditions are described in terms of a suitable directional curvature functional that allows us to derive no-gap second-order optimality conditions in an abstract setting. Our approach allows us to apply our results even when classical assumptions such as the second-order regularity conditions to the feasible set fail, extending the applicability of our approach. As applications to mathematical programming, we provide new primal and dual Karush-Kuhn-Tucker (KKT) second-order necessary and sufficient conditions. We provide some examples to illustrate our findings.

We present an alternative view for the study of optimal control of partially observed Markov Decision Processes (POMDPs). We first revisit the traditional (and by now standard) separated-design method of reducing the problem to fully observed MDPs (belief-MDPs), and present conditions for the existence of optimal policies. Then, rather than working with this standard method, we define a Markov chain taking values in an infinite dimensional product space with the history process serving as the controlled state process and a further refinement in which the control actions and the state process are causally conditionally independent given the measurement/information process. We provide new sufficient conditions for the existence of optimal control policies under the discounted cost and average cost infinite horizon criteria. In particular, while in the belief-MDP reduction of POMDPs, weak Feller condition requirement imposes total variation continuity on either the system kernel or the measurement kernel, with the approach of this paper only weak continuity of both the transition kernel and the measurement kernel is needed (and total variation continuity is not) together with regularity conditions related to filter stability. For the discounted cost setup, we establish near optimality of finite window policies via a direct argument involving near optimality of quantized approximations for MDPs under weak Feller continuity, where finite truncations of memory can be viewed as quantizations of infinite memory with a uniform diameter in each finite window restriction under the product metric. For the average cost setup, we provide new existence conditions and also a general approach on how to initialize the randomness which we show to establish convergence to optimal cost. In the control-free case, our analysis leads to new and weak conditions for the existence and uniqueness of invariant probability measures for nonlinear filter processes, where we show that unique ergodicity of the measurement process and a measurability condition related to filter stability leads to unique ergodicity.

In this paper we establish necessary optimality conditions for minimax multiprocess problems: these are optimal control problems in which we have a family of control systems coupled by endpoint constraints and a minimax cost functional to minimize.

We develop an optimal control framework that enables to determine the most-beneficial ways of investing in technology and directing capital within an economy. Our developed framework features three main novelties: the optimization of a cross–diffusion term that incorporates the allocation of capital towards specific regions with higher level of technology; the coupling of technological progress with the capital in the state system; and the inclusion of an inequality constraint imposing that the squared norm of technological progress does not surpass a capacity (M_A>0), which is more practical in economic applications. This leads to a new state-constrained optimal control problem which we analyze as follows. First, by examining the weak well-posedness of the dynamics, we identify a threshold parameter (M^*>0) such that when (M_Age M^*), the state-constraint can be omitted. In this case, we deal with a reduced state-unconstrained optimal control problem. On the other hand, when (M_A<M^*), the state-constraint is not implicitly incorporated. Consequently, we proceed by a penalization approach to formulate a sequence of state-unconstrained optimal control problems and provide necessary optimality conditions for its associated sequence of locally optimal solutions. Subsequently, we prove that the sequence of locally optimal solutions converges strongly to a locally optimal solution for the original state-constrained optimal control problem and retrieve its necessary optimality conditions. Finally, we perform various numerical simulations to illustrate the effects of optimal investment in technology and optimal capital direction on the economy. This study could offer interesting insights in the perspective of circular economy transition.

The aim of the present paper is to investigate the optimal vaccination policies with prevalence restrictions in an SIRS demographic model. We provide a well-posedness result for the system and give a thorough description of safety zones (immunity and feasible) when intensive care units (ICU) restrictions are enforced on the prevalence. Using Pontryagin’s principle for state-constrained dynamics we show that the optimal vaccination policy is of bang–bang type and give further specifics on the precise structure. The paper is intended as a counter-part to Avram et al. (Appl Math Comput 418:126816, 2022) where non-pharmaceutical interventions have been considered.

The paper is concerned with a chemotaxis model with nonlinear indirect signal consumption and density-suppressed motility

under homogeneous Neumann boundary conditions in a smooth bounded domain (Omega subset mathbb {R}^n) ((nge 1)), where the parameters (delta ), (beta >0), and (varphi (v)) is a motility function. The purpose of this paper is to determine the size of the absorption exponent to ensure the existence of global bounded classical solutions to the problem. Specifically, we first showed that when (f(u)=0), the system has a global bounded classical solution if (beta le 2), (n=1), or (beta <frac{4}{n}), (nge 2), or suitably small initial data. Subsequently, when (f(u)=ru-mu u^alpha ) with (rin mathbb {R}), (mu >0), (alpha >1), it was shown that the system admits a global bounded classical solution if (beta le alpha ), (n=1) or (beta <max bigl {alpha -1, frac{2alpha }{n}bigr }), (nge 2), and that in the critical case (beta =alpha -1), (nge 2), we proved the existence of global bounded classical solutions provided that (mu ) is properly large. Moreover, we obtained the uniform convergence of bounded solutions to the system by constructing some suitable functionals.