Optimal control of sterilization of prepackaged food in the case of problem with free final time and phase constraints

This paper is concerned with the optimal control of the sterilization of prepackaged food. The investigated system is constructed as an optimal control problem with free final horizon and phase constraints. Pontryagin’s maximum principle, the necessary optimality condition for the system, is studied by the Dubovitskii and Milyutin functional analytical approach. The derived necessary condition is presented for the problem with both the control constraints and the state constraints.     

Optimal birth control of population dynamics

The authors studied optimal birth control policies for an age-structured population of McKendrick type which is a distributed parameter system involving 1st order partial differential equations with nonlocal bilinear boundary control. The functional analytic approach of Dubovitskii and Milyutin is adopted in the investigation. Maximum principles for problems with a free end condition and fixed final horizon are developed, and the time optimal control problems, the problem with target sets, and infinite planning horizon case are investigated.     

Optimal duration and dosage of BCG intravesical immunotherapy: A free final time optimal control approach

In this paper, we propose a free final time optimal control approach applied to 4 ordinary differential equations which describe the tumor‐immune interactions after the injection of the bacillus Calmette‐Guérin (BCG) in the bladder of a hypothetical patient. The main goal of this optimal control strategy is to find the optimal dosage amount needed in each instillation of BCG for stimulating the immune‐system and then killing superficial bladder tumors and to determine the optimal duration of treatment, adequate for stopping the intravesical therapy with lesser side‐effects. For this, we introduce into the model of interest, a control function which represents the dose of BCG immunotherapy procedure and we formulate a minimization problem where the final time is considered free (nonfixed). The characterization of the sought optimal control noted u^∗ is derived based on Pontryagin's maximum principle, while the formulation of the sought optimal final time noted is based on formulae of sensitivity which are obtained conditions from the derivative of the objective function with respect to . We investigate the resolution of the free final time optimal control problem in 3 possible cases: (a) when is quadratic in the final gain function, (b) when the final gain function does not depend on , and (c) when is linear in the final gain function. Finally, we obtain the sought optimal dose of BCG, and we conclude that in case (a), we obtain an optimal duration which is more beneficial regarding the activation of immune cells while cases (b) and (c) both provide an optimal duration which is more adequate for the minimization of the tumoral population.     

具终端观测且与年龄相关的非线性种群扩散系统的最优控制

讨论了一类具终端观测且与年龄相关的非线性时变种群扩散系统的最优分布控制问题利用偏微控制理论和先验估计,证明了系统最优分布控制的存在性,得到了控制为最优的一阶必要条件,并进而讨论了系统的最优反馈控制问题.     

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

In our preceding paper [Sun B, Wu MX. Appl. Anal. 2013;92:901–921], we investigated an optimal control problem of age-structured population dynamics for spread of universally fatal diseases and derived the necessary optimality condition for the problem in fixed final horizon case. As a follow-up, in this paper, under weaker additional conditions, we address ourselves to the investigation of the foregoing system in free final horizon case and present further new results of current interests.     

Optimal control in a mathematical model for leukemia therapy with phase constraints

We examine the optimal control problem that arises in the mathematical modeling of leukemia therapy, to solve which the Pontryagin maximum principle and the penalty function method are employed. It is assumed that the drug is capable of killing not only diseased cells, but healthy cells as well. The character of the drug??s interaction with cells is described by appropriate therapy functions.     

Optimal control problem with phase constraints for a linear system of equations with delay

The optimal control problem for a linear system of differential equations with delay for which the initial function on the initial set is the control function is considered. Phase constraints with variable times are imposed on the trajectory of the object. Necessary optimality conditions of controls in admissible classes are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 12, pp. 1647–1652, December, 1991.     

Optimal control for age-structured population dynamics of incomplete data

We are concerned with the control question for linear age-structured population dynamics of incomplete initial data. More precisely, the initial population age distribution is supposed to be unknown. We here generalize the notion of no-regret control of Lions (1992) [10] to such singular population dynamics, following the method by Nakoulima, Omrane and Vélin (2000) [16]. We prove that the problem we are considering has a unique no-regret control that we characterize by a singular optimality system.     

Optimal impulsive control problems with state and mixed constraints

Doklady Mathematics -     

Stochastic control with exit time and constraints,application to small time attainability of sets

We study the existence of a solution of controlled stochastic differential equations remaining in a given set of constraints at any time smaller than the exit time of a given open set. We also investigate the small time attainability of a given closed set K, i.e., the property that, for all arbitrary small time horizon T and for all initial condition in a sufficiently small neighborhood of K, there exists a solution to the controlled stochastic differential equation which reaches K before T.     

Optimal control of damped Klein-Gordon equations with state constraints

In this paper, we study the maximum principles for optimal control problems governed by the damped Klein-Gordon equations with state constraints. And we prove the existence of the optimal parameter and deduce the necessary conditions on the optimal parameter.     

Optimal control for a population dynamics with age-dependent and diffusion in a periodic environment

Here we consider the optimal harvesting problem for a periodic age-dependent population dynamics with diffusion. Namely, we investigate the model with periodic vital rates and a periodic forcing term that sustains oscillations. By using Mazur’s theorem, we demonstrate existence of solutions of the optimal control problem and by the conception of normal cone, we also obtain the first order necessary conditions of optimality for the problem. Our results extend some known criteria.     

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.     

Numerical computation of optimal control problems with unknown final time

This paper presents a numerical computational method for the solution of optimal control problems with unknown final time. The method presented is based on a transformation and a modified quasilinearization technique. A numerical example is included for illustration.     

Optimal control in coefficients of elliptic equations with state constraints

This paper is concerned with state constrained optimal control problems of elliptic equations, the control being a coefficient of the partial differential equation. Existence of an optimal control is proved and optimality conditions are derived. We perform finite-element approximations of optimal control problems and state some convergence results: we prove convergence of optimal controls and states as well as convergence of Lagrange multipliers.This research was partially supported by the Dirección General de Investigación Científica y Técnica (Madrid).     

Optimal control of a nonlinear singular system with state constraints

A control system described by a nonlinear equation of parabolic type is considered in the situation where there may be no global solution. A particular optimal control problem subject to state constraints is studied. A proof of the existence of an optimal control is presented. The penalty method is used to obtain necessary conditions for optimal control. A proof of the convergence of this method is given. The successive approximation method is used to obtain an approximate solution for the conditions derived. Translated fromMatematicheskie Zametki, Vol. 60, No. 4, pp. 511–518, October, 1996.     

Optimal control of switched systems with time delay

In this note, we develop a computational method for solving an optimal control problem which is governed by a switched dynamical system with time delay. Our approach is to parameterize the switching instants as a new parameter vector to be optimized. Then, we derive the required gradient of the cost function which is obtained via solving a number of delay differential equations forward in time. On this basis, the optimal control problem can be solved as a mathematical programming problem.