Multipoint Necessary Optimality Conditions for Singular Controls in Processes Described by the System of Volterra Integral Equations

We consider the optimal control problem described by the system of Volterra nonlinear integral equations. The necessary optimality conditions for controls that are singular in the sense of the Pontryagin maximum principle are obtained.     

On a problem of optimal control of mobile sources

Consideration was given to the problem of optimal control of the mobile sources for systems obeying a parabolic equation together with systems of ordinary differential equations. The necessary conditions for optimality were established in the form of the pointwise and integral principles of maximum. The theoretical conclusions are illustrated by the solution of a numerical example     

Necessary first and second order optimality conditions in a discrete optimal control problem

An optimal control problem described by a system of two-dimensional nonlinear difference Volterra-type equations that are a discrete analogue of the two-dimensional integral Volterra equation is considered. The first and second variations of the quality functional are calculated assuming that the control domain is open. They are used to obtain an analogue of the Euler equation and to find the necessary second order optimality conditions that can be checked constructively.     

Necessary integral optimality conditions for quasi-singular controls in one control problem

An optimal control problem with variable structure described by a system of nonlinear integral Volterra-type equations is considered in the article. A necessary optimality condition is obtained in the form of a linearized maximum condition. A case of degeneracy of the linearized maximum principle is further studied. Integral necessary optimality conditions of quasi-singular controls are proved.     

Impulsive Synchronization of Stochastic Neural Networks via Controlling Partial States

In this paper the perceptron neural networks are applied to approximate the solution of fractional optimal control problems. The necessary (and also sufficient in most cases) optimality conditions are stated in a form of fractional two-point boundary value problem. Then this problem is converted to a Volterra integral equation. By using perceptron neural network’s ability in approximating a nonlinear function, first we propose approximating functions to estimate control, state and co-state functions which they satisfy the initial or boundary conditions. The approximating functions contain neural network with unknown weights. Using an optimization approach, the weights are adjusted such that the approximating functions satisfy the optimality conditions of fractional optimal control problem. Numerical results illustrate the advantages of the method.     

Optimal control of mobile sources for heat conductivity processes

We consider a problem of optimal control of the mobile sources for heat conductivity processes, described by parabolic equation and systems of ordinary differential equations. Sufficient conditions for Frechet differentiability of the performance criterion and an expression for its gradient were determined. The necessary optimality condition was established in the form of the integral principles of maximum. The theoretical conclusions are illustrated by the solution of a numerical example.     

Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria

Optimal control of general nonlinear nonaffine controlled systems with nonquadratic performance criteria (that permit state- and control-dependent time-varying weighting parameters), is solved classically using a sequence of linear- quadratic and time-varying problems. The proposed method introduces an “approximating sequence of Riccati equations” (ASRE) to explicitly construct nonlinear time-varying optimal state-feedback controllers for such nonlinear systems. Under very mild conditions of local Lipschitz continuity, the sequences converge (globally) to nonlinear optimal stabilizing feedback controls. The computational simplicity and effectiveness of the ASRE algorithm is an appealing alternative to the tedious and laborious task of solving the Hamilton–Jacobi–Bellman partial differential equation. So the optimality of the ASRE control is studied by considering the original nonlinear-nonquadratic optimization problem and the corresponding necessary conditions for optimality, derived from Pontryagin's maximum principle. Global optimal stabilizing state-feedback control laws are then constructed. This is compared with the optimality of the ASRE control by considering a nonlinear fighter aircraft control system, which is nonaffine in the control. Numerical simulations are used to illustrate the application of the ASRE methodology, which demonstrate its superior performance and optimality.     

Lipschitzian Regularity of the Minimizing Trajectories for Nonlinear Optimal Control Problems

We consider the Lagrange problem of optimal control with unrestricted controls and address the question: under what conditions can we assure optimal controls are bounded? This question is related to one of Lipschitzian regularity of optimal trajectories, and the answer to it is crucial in closing the gap between the conditions arising in existence theory and necessary optimality conditions. Rewriting the Lagrange problem in a parametric form, we obtain a relation between the applicability conditions of the Pontryagin maximum principle to the latter problem and the Lipschitzian regularity conditions for the original problem. Under the standard hypotheses of coercivity of the existence theory, the conditions imply that the optimal controls are essentially bounded, assuring the applicability of the classical necessary optimality conditions like the Pontryagin maximum principle. The result extends previous Lipschitzian regularity results to cover optimal control problems with general nonlinear dynamics.     

Application of abstract variational theory to hereditary systems--A survey

In this survey paper, results for optimal control problems governed by hereditary systems are presented and discussed. Although fundamental results (controllability, existence and uniqueness of optimal controls, feedback controls) obtained without employing abstract variational approaches are briefly reviewed, emphasis is placed on those results established through the use of general abstract variational theories, and, in particular, on those growing out of the theories developed by the late Lucien W. Neustadt. Motivating examples of hereditary systems encountered in applications (technological, biological, and biomedical) are given. Abstract variational theories involving the Lagrange Multiplier rule and the Kuhn-Tucker conditions in a Banach space, the approach of Dubovitskii-Milyutin, the quasiconvexity ideas of Gamkrelidze, and the abstract maximum principles developed in the work of Halkin and Neustadt are summarized briefly. A more detailed discussion of applications of the quasiconvexity ideas and the Neustadt theory to systems governed by differential-difference, functional-differential, and Volterra integral equations is presented for control problems both with and without state constraints. Recent results for control of hereditary systems with terminal function space boundary conditions are also reviewed.     

Maximizing a state convex lagrange functional in optimal control

We consider the maximization problem for an integral functional with a state-convex integrand function along a standard control system. We show necessary and sufficient global optimality conditions related to the Pontryagin’s maximum principle. We study the properties of these conditions and their relations with optimal control theory. We also illustrate the efficiency of the resulting conditions on specific examples.     

On optimal control problem for moving sources for a parabolic equation

A variational method for solving an optimal control problem for moving sources for systems, their states described by a parabolic-type equation, is considered. The necessary optimality conditions are found in the form of pointwise and integral maximum principles. The theoretical conclusions are illustrated by a numerical example.     

Nonconvex optimal control of nonlinear monotone parabolic systems

An optimal control problem is studied for distributed systems governed by nonlinear parabolic PDE's with state constraints. The state equation is monotone in the state variable and nonlinear in the control variable. The constraints and the cost functional are not necessarily convex. Relaxed controls are used to prove the existence of an optimal control. Moreover, a minimum principle of relaxed optimality is established.     

Optimal control with partially specified input functions†

This paper advances some aspects of the extended maximum principle for systems with partially specified control functions. The end-time condition, which has been ignored in previous works, is derived in this paper. An alternative form of the condition of the optimality which has been overlooked in the previous papers is also presented. It is shown that, in contrast to the maximum principle of Pontryagin, this extended maximum principle states that a necessary condition for a control-and-path pair to be optimal is that the time integral of the Hamiltonian function, not the Hamiltonian function itself, attains its maximum. If the end time of the control process is free, then an additional necessary condition is that the Hamiltonian function must cross over the time axis at the end time from above. These conditions may he combined to state that the time integral of the Hamiltonian functions attains its maximum with respect to the partial variations of both the control input and the end time

The usefulness of the conditions of optimality derived in this paper is illustrated in an example of optimal control problem which does not fall within the framework of Pontryagin's maximum principle. The control system used in the example is subject to piecewise constant control functions with two switchings pre-specified at one-third and two-thirds of the total time of control operation, respectively, with the weighted sum of control time and energy consumption as the performance criterion     

Positional solutions of Hamilton-Jacobi equations in control problems for discrete-continuous systems

We develop a canonical global optimality theory based on operating with the set of solutions for the Hamilton-Jacobi inequalities that parametrically depend on the initial (or final) position. These solutions, called positional L-functions (of Lyapunov type), naturally arise in the studies of control problems for discrete-continuous (hybrid, impulse) systems; an important prototype of such problems are classical optimal control problems with general end constraints on the trajectory. We analyze sufficient optimality conditions with this new class of L-functions and invert the maximum principle into a sufficient condition for nonlinear problems of optimal impulse control.     

Functional expansions for the response of nonlinear differential systems

This paper is concerned with representing the response of nonlinear differential systems by functional expansions. An abstract theory of variational expansions, similar to that of L. M. Graves (1927), is developed. It leads directly to concrete expressions (multilinear integral operators) for the functionals of the expansions and sets conditions on the differential systems which insure that the expansions give reasonable approximations of the response. Similarly, it is shown that the theory of analytic functions in Banach spaces leads directly to conditions which imply uniform convergence of functional series. The main results on differential systems are summarized in a set of theorems, some of which overlap and extend the recent results of Brockett on Volterra series representations for the response of linear analytic differential systems. Other theorems apply to more general nonlinear differential systems. They provide a rigorous foundation for a large body of previous research on Volterra series expansions. The multilinear integral operators are obtained from systems of differential equations which characterize exactly the variations. These equations are of much lower order than those obtained by the technique of Carleman. A nonlinear feedback system serves as an example of an application of the theory.     

High-order necessary conditions of optimality for nonlinear control systems

For nonlinear affine control systems with unbounded controls, necessary high-order optimality conditions are derived. These conditions are stated in the form of a high-order maximum principle and are expressed in terms of Lie brackets and Newton diagrams.     

A constrained optimal control of nonlinear systems

A new method for optimal control of nonlinear systems with input constraint is discussed in this paper. The system is optimized by minimizing a quadratic performance index. Considering this problem as a nonlinear programming problem, the necessary and sufficient conditions for optimal control are derived, which are later simplified to an integral equation. This integral equation becomes a necessary condition. The existence of a solution of this integral equation is studied, and a method of solving it is discussed. A numerical example is worked out at the end.     

带Markov跳的离散时间随机控制系统的最大值原理

本文研究一类同时含有Markov跳过程和乘性噪声的离散时间非线性随机系统的最优控制问题, 给出并证明了相应的最大值原理. 首先, 利用条件期望的平滑性, 通过引入具有适应解的倒向随机差分方程, 给出了带有线性差分方程约束的线性泛函的表示形式, 并利用Riesz定理证明其唯一性. 其次, 对带Markov跳的非线性随机控制系统, 利用针状变分法, 对状态方程进行一阶变分, 获得其变分所满足的线性差分方程. 然后, 在引入Hamilton函数的基础上, 通过一对由倒向随机差分方程刻画的伴随方程, 给出并证明了带有Markov跳的离散时间非线性随机最优控制问题的最大值原理, 并给出该最优控制问题的一个充分条件和相应的Hamilton-Jacobi-Bellman方程. 最后, 通过 一个实际例子说明了所提理论的实用性和可行性.     

Robust maximum principle for multi-model LQ-problem

This paper presents the version of the robust maximum principle in the context of multi-model control formulated as the minimax Bolza problem. The cost function contains a terminal term as well as an integral one. A fixed horizon and terminal set are considered. The necessary conditions of the optimality are derived for the class of uncertain systems given by an ordinary differential equation with parameters from a given finite set. This problem consists in the control design providing a good behaviour for a given class of multi-model system. It is shown that the design of the minimax optimal controller is reduced to a finite-dimensional optimization problem given at the corresponding simplex set containing the weight parameters to be found. The robust optimal control may be interpreted as a mixture (with the optimal weights) of the controls which are optimal for each fixed parameter value. The proof is based on the recent results obtained for minimax Mayer problem (Boltyanski and Poznyak 1999a). The minimax linear quadratic control problem is considered in detail and the illustrative examples dealing with finite as well as infinite horizons conclude this paper.     

Approximate optimal controls for a class of Volterra systems with nonlinear delay

Necessary and sufficient conditions of approximate optimal controls for integral linear in state Volterra systems with nonlinear delays are derived. They are used for studying approximate optimal controls obtained by the control parameterization method as an important application.