Optimality conditions for age-structured control systems

We consider a fairly general model (extension of the Gurtin-MacCamy model of population dynamics) of an age structured control system with nonlocal dynamics and nonlocal boundary conditions. A necessary optimality condition is obtained in the form of Pontryagin's maximum principle, which is applicable to a number of practically meaningful models where the previously known results fail. We discuss such models (an epidemic control, and a capital accumulation model) as illustrations.     

Optimal control: Nonlocal conditions,computational methods,and the variational principle of maximum

This paper surveys theoretical results on the Pontryagin maximum principle (together with its conversion) and nonlocal optimality conditions based on the use of the Lyapunov-type functions (solutions to the Hamilton-Jacobi inequalities). We pay special attention to the conversion of the maximum principle to a sufficient condition for the global and strong minimum without assumptions of the linear convexity, normality, or controllability. We give the survey of computational methods for solving classical optimal control problems and describe nonstandard procedures of nonlocal improvement of admissible processes in linear and quadratic problems. Furthermore, we cite some recent results on the variational principle of maximum in hyperbolic control systems. This principle is the strongest first order necessary optimality condition; it implies the classical maximum principle as a consequence.     

Adaptive event-triggered dynamic distributed control of switched positive systems with switching faults

This paper designs the dynamic output-feedback controller of switched positive systems subject to switching faults using an improved adaptive event-triggering mechanism. An adaptive event-triggering condition is addressed in the form of 1-norm by virtue of the measurable outputs of distributed sensors and the corresponding error. An error-based closed-loop control system whose dynamic variable relies on a state observer is obtained. A multiple copositive Lyapunov function is constructed to deal with the positivity and stability of the systems. The matrix decomposition and linear programming approaches are used to design and compute the controller and observer gains. An improved average dwell time scheme is proposed to handle the switching faults. The contributions of this paper lie in that: (i) An adaptive event-triggering mechanism is established for switched positive systems, (ii) A framework on the fault of switching signal is constructed, and (iii) A dynamic distributed controller is proposed for the considered systems. Finally, two illustrative examples are given to verify the effectiveness of the obtained results.     

Method for nonlocal improvement of extreme controls in the maximization of the terminal state norm

A numerical approach is applied to the maximization of a positive definite quadratic form on the set of terminal states of a linear system with interval constraints on the control. An optimality criterion is used to develop a method for nonlocal improvement of controls satisfying the maximum principle (extreme points of the reachable set). The iterative procedure of the method is proved to converge. Numerical results are presented.     

Numerical Solution of a Scalar One-Dimensional Monotonicity-Preserving Nonlocal Nonlinear Conservation Law

In this paper, we present numerical studies of a recently proposed scalar nonlocal nonlinear conservation law in one space dimension. The nonlocal model accounts for nonlocal interactions over a finite horizon and enjoys maximum principle, monotonicity-preserving and entropy condition on the continuum level. Moreover, it has a well-defined local limit given by a conventional local conservation laws in the form of partial differential equations. We discuss convergent numerical approximations that preserve similar properties on the discrete level. We also present numerical experiments to study various limiting behavior of the numerical solutions.     

On discrete maximum principle

A model of optimal control for discrete systems and the historical development of the discrete maximum principle are considered. The paper deals with local optimality conditions of the first order, e.g. with a local maximum-principle and a quasi-maximum principle. Furthermore, optimality conditions of higher order, e. g. an optimality condition for singular controls are given.     

Optimal control of time-delay systems with distributed parameters

An optimal control problem with a prescribed performance index for parabolic systems with time delays is investigated. A necessary condition for optimality is formulated and proved in the form of a maximum principle. Under additional conditions, the maximum principle gives sufficient conditions for optimality. It is also shown that the optimal control is unique. As an illustration of the theoretical consideration, an analytic solution is obtained for a time-delayed diffusion system.The author wishes to express his deep gratitude to Professors J. M. Sloss and S. Adali for the valuable guidance and constant encouragement during the preparation of this paper.     

On the maximum principle for a class of discrete dynamical systems with lags

A maximum principle for a class of discrete systems with lags is established constructively within the framework of discrete optimal control theory. An application of the maximum principle to a problem in advertising concludes the paper.     

Extremality,Controllability, and Abundant Subsets of Generalized Control Systems

The generalized control system that we consider in this paper is a collection of vector fields, which are measurable in the time variable and Lipschitzian in the state variable. For such system, we define the concept of an abundant subset. Our definition follows the definition of an abundant set of control functions introduced by Warga. We prove a controllability–extremality theorem for generalized control systems, which says, in essence, that either a given trajectory satisfies a type of maximum principle or a neighborhood of the endpoint of the trajectory can be covered by trajectories of an abundant subset. We apply the theorem to a control system in the classical formulation and obtain a controllability–extremality result, which is stronger, in some respects, than all previous results of this type. Finally, we apply the theorem to differential inclusions and obtain, as an easy corollary, a Pontryagin-type maximum principle for nonconvex inclusions.     

Pontryagin’s maximum principle for dynamic systems on time scales

In this work, an analogue of Pontryagin’s maximum principle for dynamic equations on time scales is given, combining the continuous and the discrete Pontryagin maximum principles and extending them to other cases ‘in between’. We generalize known results to the case when a certain set of admissible values of the control is not necessarily closed (but convex) and the attainable set is not necessarily convex. At the same time, we impose an additional condition on the graininess of the time scale. For linear systems, sufficient conditions in the form of the maximum principle are obtained.     

Optimal control of age-structured population dynamics for spread of universally fatal diseases

This article is concerned with the optimal control problem of age-structured population dynamics for the spread of universally fatal diseases. The existence and uniqueness of solution of the system, which consists of a group of partial differential equations with nonlocal boundary conditions, is proved. The Dubovitskii and Milyutin functional analytical approach is adopted in the investigation of the Pontryagin maximum principle of the system. The necessary condition is presented for the optimal control problem in fixed final horizon case. Finally, a remark on how to utilize the obtained results is also made.     

Iterative learning consensus control with initial state learning for fractional order distributed parameter models multi-agent systems

This paper considers the consensus control problem of multi-agent systems (MAS) with distributed parameter models. Based on the framework of network topologies, a second-order PI-type iterative learning control (ILC) protocol with initial state learning is proposed by using the nearest neighbor knowledge. A discrete system for proposed ILC is established, and the consensus control problem is then converted to a stability problem for such a discrete system. Furthermore, by using generalized Gronwall inequality, a sufficient condition for the convergence of the consensus errors between any two agents is obtained. Finally, the validity of the proposed method is verified by two numerical examples.     

Controlled jump processes

Finite and infinite planning horizon Markov decision problems are formulated for a class of jump processes with general state and action spaces and controls which are measurable functions on the time axis taking values in an appropriate metrizable vector space. For the finite horizon problem, the maximum expected reward is the unique solution, which exists, of a certain differential equation and is a strongly continuous function in the space of upper semi-continuous functions. A necessary and sufficient condition is provided for an admissible control to be optimal, and a sufficient condition is provided for the existence of a measurable optimal policy. For the infinite horizon problem, the maximum expected total reward is the fixed point of a certain operator on the space of upper semi-continuous functions. A stationary policy is optimal over all measurable policies in the transient and discounted cases as well as, with certain added conditions, in the positive and negative cases.     

Optimal birth control of population dynamics. II. Problems with free final time,phase constraints,and mini-max costs

A previous analysis of optimal birth control of population systems of the McKendrick type (a distributed parameter system involving 1st order partial differential equations with nonlocal bilinear boundary control) raised 3 additional issues--free final time problem, system with phase constraints, and the mini-max control problem of a population. The free final time problem considers the minimum time problem to be a special case, but relaxes many convexity assumptions. Theorems (maximum principles) and corollaries are developed that flow from the terminology and mathematical notations set forth in the earlier article.     

Numerical analysis and applications of Fokker-Planck equations for stochastic dynamical systems with multiplicative α-stable noises

In this paper, we study the nonlocal Fokker-Planck equations (FPEs) associated with Lévy-driven scalar stochastic dynamical systems. We first derive the Fokker-Planck equation for the case of multiplicative symmetric α-stable noises, by the adjoint operator method. Then we construct a finite difference scheme to simulate the nonlocal FPE on either bounded or infinite domain. It is shown that the semi-discrete scheme satisfies the discrete maximum principle and converges. Some experiments are conducted to validate the numerical method. Finally, we extend the results to the asymmetric case and present an application to the nonlinear filtering problem.     

Optimal Control of Heterogeneous Systems with Endogenous Domain of Heterogeneity

The paper deals with optimal control of heterogeneous systems, that is, families of controlled ODEs parameterized by a parameter running over a domain called domain of heterogeneity. The main novelty in the paper is that the domain of heterogeneity is endogenous: it may depend on the control and on the state of the system. This extension is crucial for several economic applications and turns out to rise interesting mathematical problems. A necessary optimality condition is derived, where one of the adjoint variables satisfies a differential inclusion (instead of equation) and the maximization of the Hamiltonian takes the form of ??min-max??. As a consequence, a Pontryagin-type maximum principle is obtained under certain regularity conditions for the optimal control. A formula for the derivative of the objective function with respect to the control from L _?? is presented together with a sufficient condition for its existence. A stylized economic example is investigated analytically and numerically.     

A RISK-SENSITIVE STOCHASTIC MAXIMUM PRINCIPLE FOR OPTIMAL CONTROL OF JUMP DIFFUSIONS AND ITS APPLICATIONS

A stochastic maximum principle for the risk-sensitive optimal control problem of jump diffusion processes with an exponential-of-integral cost functional is derived assuming that the value function is smooth, where the diffusion and jump term may both depend on the control. The form of the maximum principle is similar to its risk-neutral counterpart. But the adjoint equations and the maximum condition heavily depend on the risk-sensitive parameter. As applications, a linear-quadratic risk-sensitive control problem is solved by using the maximum principle derived and explicit optimal control is obtained.     

Isothermic Surfaces in Spaces of Arbitrary Dimension: Integrability, Discretization, and Bäcklund Transformations—A Discrete Calapso Equation

A vector analog of the classical Calapso equation governing isothermic surfaces in R ^{n +2} is introduced. It is shown that this vector Calapso system admits a nonlocal) scalar Lax pair based on the classical Moutard equation. The analog of Darboux's Bäcklund transformation for isothermic surfaces in R³ is derived in a systematic manner and shown that it may be formulated in terms of the classical Moutard transformation acting on the scalar Lax pair. A permutability theorem for isothermic surfaces is set down that manifests itself in an explicit superposition principle for the vector Calapso system. This superposition principle in vectorial form is shown to constitute an integrable discretization of the vector Calapso system and, therefore, defines discrete isothermic surfaces in R ^{n +2}. The discrete Calapso equation is related to the discrete Korteweg–de Vries equation and discrete holomorphic functions. A matrix Lax pair based on Clifford algebras and a scalar Lax pair are derived for the discrete Calapso equation. A discrete Moutard-type transformation for the discrete Calapso equation is obtained, and it is shown that the discrete Calapso equation may be specialized to an integrable discrete version of the O( n +2) nonlinear σ-model.     

Maximum principle for optimal distributed control of the viscous Dullin-Gottwald-Holm equation

In this paper, the optimal distributed control of the viscous Dullin-Gottwald-Holm equation is investigated. Adopting the Dubovitskii and Milyutin functional analytical approach, we obtain the Pontryagin maximum principle of the system. The necessary optimality condition is established for an optimal control problem in fixed final horizon case. Finally, an illustrative example is also given.     

非脆弱离散重复控制的二维模型方法

针对一类线性离散系统,提出一种基于二维模型的非脆弱离散重复控制设计方法.通过独立地考虑重复控制系统的控制与学习行为,建立离散重复控制系统的二维模型. 在此基础上,针对重复控制器和反馈控制器具有不确定性的离散重复控制系统,给出了基于线性矩阵不等式的系统稳定性条件和重复控制律. 最后,数值仿真实例验证了所提方法的有效性.