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We analyze a class of nonlinear partial differential equations (PDEs) defined on

We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body

This paper focuses on the role of a government of a large population of interacting agents as a meanfield optimal control problem derived from deterministic finite agent dynamics. The control problems are constrained by a Partial Differential Equation of continuity-type without diffusion, governing the dynamics of the probability distribution of the agent population. We derive existence of optimal controls in a measure-theoretical setting as natural limits of finite agent optimal controls without any assumption on the regularity of control competitors. In particular, we prove the consistency of mean-field optimal controls with corresponding underlying finite agent ones. The results follow from a Γ -convergence argument constructed over the mean-field limit, which stems from leveraging the superposition principle.

In this paper, a novel two-stage epidemic model with a dynamic control strategy is proposed to describe the spread of Corona Virus Disease 2019 (COVID-19) in China. Combined with local epidemic control policies, an epidemic model with a traceability process is established. We aim to investigate the appropriate control strategies to minimize the control cost and ensure the normal operation of society under the premise of containing the epidemic. This work mainly includes: (i) propose the concept about the first and the second waves of COVID-19, as well as study the case data and regularity of four cities; (ii) derive the existence and stability of the equilibrium, the parameter sensitivity of the model, and the existence of the optimal control strategy; (iii) carry out the numerical simulation associated with the theoretical results and construct a dynamic control strategy and verify its feasibility.

In this article, we derive first-order necessary optimality conditions for a constrained optimal control problem formulated in the Wasserstein space of probability measures. To this end, we introduce a new notion of localised metric subdifferential for compactly supported probability measures, and investigate the intrinsic linearised Cauchy problems associated to non-local continuity equations. In particular, we show that when the velocity perturbations belong to the tangent cone to the convexification of the set of admissible velocities, the solutions of these linearised problems are tangent to the solution set of the corresponding continuity inclusion. We then make use of these novel concepts to provide a synthetic and geometric proof of the celebrated Pontryagin Maximum Principle for an optimal control problem with inequality final-point constraints. In addition, we propose sufficient conditions ensuring the normality of the maximum principle.

The idea of optimization can be regarded as an important basis of many disciplines and hence is extremely useful for a large number of research fields, particularly for artificial-intelligence-based advanced control design. Due to the difficulty of solving optimal control problems for general nonlinear systems, it is necessary to establish a kind of novel learning strategies with intelligent components. Besides, the rapid development of computer and networked techniques promotes the research on optimal control within discrete-time domain. In this paper, the bases, the derivation, and recent progresses of critic intelligence for discrete-time advanced optimal control design are presented with an emphasis on the iterative framework. Among them, the so-called critic intelligence methodology is highlighted, which integrates learning approximators and the reinforcement formulation.

We derive a maximum principle for optimal control problems with constraints given by the coupling of a system of ordinary differential equations and a partial differential equation of Vlasov type with smooth interaction kernel. Such problems arise naturally as Gamma-limits of optimal control problems constrained by ordinary differential equations, modeling, for instance, external interventions on crowd dynamics by means of leaders. We obtain these first-order optimality conditions in the form of Hamiltonian flows in the Wasserstein space of probability measures with forward–backward boundary conditions with respect to the first and second marginals, respectively. In particular, we recover the equations and their solutions by means of a constructive procedure, which can be seen as the mean-field limit of the Pontryagin Maximum Principle applied to the optimal control problem for the discretized density, under a suitable scaling of the adjoint variables.

We take a new look at the relation between the optimal transport problem and the Schrödinger bridge problem from a stochastic control perspective. Our aim is to highlight new connections between the two that are richer and deeper than those previously described in the literature. We begin with an elementary derivation of the Benamou–Brenier fluid dynamic version of the optimal transport problem and provide, in parallel, a new fluid dynamic version of the Schrödinger bridge problem. We observe that the latter establishes an important connection with optimal transport without zero-noise limits and solves a question posed by Eric Carlen in 2006. Indeed, the two variational problems differ by a Fisher information functional . We motivate and consider a generalization of optimal mass transport in the form of a (fluid dynamic) problem of optimal transport with prior . This can be seen as the zero-noise limit of Schrödinger bridges when the prior is any Markovian evolution. We finally specialize to the Gaussian case and derive an explicit computational theory based on matrix Riccati differential equations. A numerical example involving Brownian particles is also provided.

In this paper, a numerical method is developed for solving a class of delay fractional optimal control problems involving nonlinear time-delay systems and subject to state inequality constraints. The fractional derivatives in this class of problems are described in the sense of Caputo, and they can be of different orders. First, we propose a numerical integration scheme for the fractional time-delay system and prove that the convergence rate of the numerical solution to the exact one is of second order based on Taylor expansion and linear interpolation. This gives rise to a discrete-time optimal control problem. Then, we derive the gradient formulas of the cost and constraint functions with respect to the decision variables and present a gradient computation procedure. On this basis, a gradient-based optimization algorithm is developed to solve the resulting discrete-time optimal control problem. Finally, several example problems are solved to demonstrate the effectiveness of the developed solution approach.

Spaced repetition is a technique for efficient memorization which uses repeated review of content following a schedule determined by a spaced repetition algorithm to improve long-term retention. However, current spaced repetition algorithms are simple rule-based heuristics with a few hard-coded parameters. Here, we introduce a flexible representation of spaced repetition using the framework of marked temporal point processes and then address the design of spaced repetition algorithms with provable guarantees as an optimal control problem for stochastic differential equations with jumps. For two well-known human memory models, we show that, if the learner aims to maximize recall probability of the content to be learned subject to a cost on the reviewing frequency, the optimal reviewing schedule is given by the recall probability itself. As a result, we can then develop a simple, scalable online spaced repetition algorithm, MEMORIZE, to determine the optimal reviewing times. We perform a large-scale natural experiment using data from Duolingo, a popular language-learning online platform, and show that learners who follow a reviewing schedule determined by our algorithm memorize more effectively than learners who follow alternative schedules determined by several heuristics.