On the optimal control of linear systems with linear performance criterion and constraints

We suggest an analytical-numerical method for solving a boundary value optimal control problem with state, integral, and control constraints. The embedding principle underlying the method is based on the general solution of a Fredholm integral equation of the first kind and its analytic representation; the method permits one to reduce the boundary value optimal control problem with constraints to an optimization problem with free right end of the trajectory.     

Asymptotics of a solution to an optimal boundary control problem

A problem of optimal boundary control of solutions to an elliptic-type equation with a small coefficient at the highest derivative and integral constraints on the control is considered. Asymptotic estimates for solutions to a problem that approximates the original problem are obtained.     

Second order optimality conditions for a class of control problems governed by non-linear integral equations with application to parabolic boundary control

In the paper necessary and sufficient second order optimality conditions for optimal control problems governed by weakly singular non linear Hammerstein integral equations are derived. They are applied to a semilinear parabolic boundary control problem for the one dimensional heat equation.     

Feedback control in LQCP with a terminal inequality constraint

This paper considers the linear-quadratic control problem (LQCP) for systems defined by evolution operators with a terminal state inequality constraint. It is shown that, under suitable assumptions, the optimal control exists, is unique, and has a closed-loop structure. The synthesis of the feedback control requires one to solve the integral Riccati equation for the unconstrainted LQCP and a linear integral equation whose solution depends on a real parameter satisfying an additional condition.This work was completed while the author was visiting the Control Theory Centre, University of Warwick, Coventry, England.     

Relaxation in nonconvex optimal control problems described by fractional differential equations

We consider the minimization problem of an integral functional with integrand that is not convex in the control on solutions of a control system described by fractional differential equation with mixed nonconvex constraints on the control. A relaxation problem is treated along with the original problem. It is proved that, under general assumptions, the relaxation problem has an optimal solution, and that for each optimal solution there is a minimizing sequence of the original problem that converges to the optimal solution with respect to the trajectory, the control, and the functional in appropriate topologies simultaneously.     

Necessary optimality conditions in an optimal control problem for a parabolic equation with nonlocal integral conditions

Doklady Mathematics - A variational method for the optimal control of moving sources governed by a parabolic equation with nonlocal integral conditions is considered. For this problem, an existence...     

Solution of elliptic optimal control problem with pointwise and non-local state constraints

We study an optimal control problem of a system governed by a linear elliptic equation, with pointwise control constraints and pointwise and non-local (integral) state constraints. We construct a finite-difference approximation of the problem, we prove the existence and the convergence of the approximate solutions to the exact solution. We construct and study mesh saddle point problem and its iterative solution method and analyze the results of numerical experiments.     

A semi-analytical direct optimal control solution for strongly excited and dissipative Hamiltonian systems

A semi-analytical direct optimal control solution for strongly excited and dissipative Hamiltonian systems is proposed based on the extended Hamiltonian principle, the Hamilton-Jacobi-Bellman (HJB) equation and its variational integral equation, and the finite time element approximation. The differential extended Hamiltonian equations for structural vibration systems are replaced by the variational integral equation, which can preserve intrinsic system structure. The optimal control law dependent on the value function is determined by the HJB equation so as to satisfy the overall optimality principle. The partial differential equation for the value function is converted into the integral equation with variational weighting. Then the successive solution of optimal control with system state is designed. The two variational integral equations are applied to sequential time elements and transformed into the algebraic equations by using the finite time element approximation. The direct optimal control on each time element is obtained respectively by solving the algebraic equations, which is unconstrained by the system state observed. The proposed control algorithm is applicable to linear and nonlinear systems with the quadratic performance index, and takes into account the effects of external excitations measured on control. Numerical examples are given to illustrate the optimal control effectiveness.     

Time-optimal control of a parabolic system with time lags given in the integral form

In this paper the time-optimal control problem for a distributedparabolic system in which time lags appear in the integral formboth in the state equation and in the boundary condition ispresented. The particular properties of the optimal controlare discussed.     

Constrained control of a nonlinear two point boundary value problem,I

In this paper we consider an optimal control problem for a nonlinear second order ordinary differential equation with integral constraints. A necessary optimality condition in form of the Pontryagin minimum principle is derived. The proof is based on McShane-variations of the optimal control, a thorough study of their behaviour in dependence of some denning parameters, a generalized Green formula for second order ordinary differential equations with measurable coefficients and certain tools of convex analysis.Dedicated to Lothar von Wolfersdorf on the occasion of his 60th birthday     

Asymptotics of the solution in a problem of optimal boundary control of a flow through a part of the boundary

We consider a problem of optimal control through a part of the boundary of solutions to an elliptic equation in a bounded domain with smooth boundary with a small parameter at the Laplace operator and integral constraints on the control. A complete asymptotic expansion of the solution to this problems in powers of the small parameter is constructed.     

Solution of a boundary control problem for a nonlinear parabolic equation

An optimal boundary control problem for a nonlinear parabolic equation of second order is considered. An existence theorem is proved and necessary optimality conditions are obtained in the form of point and integral maximum principles.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 30–37, 1986.     

Optimal control of piecewise deterministic markov process

The trajectories of piecewise deterministic Markov processes are solutions of an ordinary (vector)differential equation with possible random jumps between the different integral curves. Both continuous deterministic motion and the random jumps of the processes are controlled in order to minimize the expected value of a performance functional consisting of continuous, jump and terminal costs. A limiting form of the Hamilton-Jacobi-Bellman partial differential equation is shown to be a necessary and sufficient optimality condition. The existence of an optimal strategy is proved and acharacterization of the value function as supremum of smooth subsolutions is also given. The approach consists of imbedding the original control problem tightly in a convex mathematical programming problem on the space of measures and then solving the latter by dualit     

Minimizing the natural fuel requirement for nuclear reactor power systems: A nonstandard optimal control problem

A class of model problems in nuclear reactor economics is defined and shown to be equivalent to a linear optimal control problem to which present versions of the maximum principle apparently cannot be applied. It is shown that the search for an optimal control can be restricted tocontrols of maximum fuel utilization (Comfu), and that theComfu's are in a one-to-one correspondence with the functions which satisfy certain inequalities and are solutions of a nonlinear Volterra integral equation containing the value of the cost functional as a parameter. In the general case, one can establish an iterative procedure, involving solution of the integral equation at each iteration, for approximating the optimalComfu. For some important special cases, a point on the solution corresponding to the optimalComfu is knowna priori, and thus the optimalComfu can be obtained by solving the integral equation only once. Some possible generalizations of the original economic model are also discussed.This research was sponsored by the US Atomic Energy Commission under contract with the Union Carbide Corporation.     

Imbedding methods for the integral equations for the optimal control of time-lag systems,I: Parameter dependence

The problem of the optimal control of systems with time lags can be reduced to a Fredholm integral equation. In this paper, it is shown that the Fredholm resolvent satisfies an initial-value problem in which a parameter serves as the independent variable. This reduction may be of computational utility.This work was partially supported by Sigma Delta Epsilon—Graduate Women in Science and NIH Grant No. GM-23732.     

Sensitivity,controllability, and necessary conditions of optimal control problems governed by integral equations

Infinite-dimensional perturbations in all constraints of an optimal control problem governed by a Volterra integral equation with the presence of a state constraint are considered. These perturbations give rise to a value function, whose analysis through the proximal normal technique provides sensitivity, controllability, and even necessary conditions for the basic problem. Actually all information about the value function is contained in Clarke's normal cone of its epigraph, which can be characterized by the proximal normal formula.     

On the optimal control of systems governed by nonlinear volterra equations

Existence theorems are stated and proved for optimal control problems monitored by a nonlinear Volterra-type integral equation. Growth conditions are taken into consideration which ensure the needed compactness on the set of trajectories and entail the existence of the usual optimal solutions for the given problem. Weak or approximate solutions are taken into consideration in a subsequent paper.The author would like to express his appreciation to Professor L. Cesari and Professor M. B. Suryanarayana for many helpful conversations and suggestions for the improvement of earlier versions of this paper.     

非定常资产发展方程的最优控制分析

讨论了一类非定常非线性的资产发展方程中资产积累率的最优控制问题,利用泛函分析和积分方程理论,得到了最优解的存在唯一性.     

On a class of optimal control problems with distributed and lumped parameters

The optimal control of moving sources governed by a parabolic equation and a system of ordinary differential equations with initial and boundary conditions is considered. For this problem, an existence and uniqueness theorem is proved, sufficient conditions for the Fréchet differentiability of the cost functional are established, an expression for its gradient is derived, and necessary optimality conditions in the form of pointwise and integral maximum principles are obtained.     

Optimal control of random evolutions

We consider a stochastic control problem for a random evolution. We study the Bellman equation of the problem and we prove the existence of an optimal stochastic control which is Markovian. This problem enables us to approximate the general problem of the optimal control of solutions of stochastic differential equations.