Stability criteria for set control differential equations

The existence and comparison results on solutions of set control differential equation were studied in [N.D. Phu, T.T. Tung, Some results on sheaf-solutions of sheaf set control problems, Nonlinear Analysis 67 (2007) 1309–1315]. In this paper, we present the stability criteria for solutions of set control differential equation.     

Some remarks on the numerical solution of bang-bang type optimal control problems

This paper is concerned with the numerical solution of optimal control problems for which each optimal control is bang-bang. Especially, the results apply to parabolic boundary control Problems. Starting from a sequence of feasible solutions converging to an optimal control u, a sequence of bang-bang controls converging to u is constructed. Bang-bang approximations of u are desirable for certain numerical reasons. Sequences of arbitrary feasible controls converging to u may be obtained by discretization or by a descent method. Numerical examples are also given.     

Some equivalent problems in set optimization

Given a set-valued optimization problem (P), there is more than one way of defining the solutions associated with it. Depending on the decision maker’s preference, we consider the vector criterion or the set criterion. Both criteria of solution are considered together to solve problem (P) by reducing the feasible set.     

Necessary conditions for optimality for control problems with time delays appearing in both state and control variables

An integral maximum principle is developed for a class of nonlinear systems containing time delays in state and control variables. Its proof is based on the theory of quasiconvex families of functions, originally developed by Gamkrelidze and extended by Banks. This result is used to obtain a pointwise principle of the Pontryagin type.The authors wish to acknowledge Professor J. M. Blatt for suggesting this problem. Further, they also wish to acknowledge the referee of the paper for bringing to their attention the problems discussed in Section 6.     

Second-order optimality principle for singular optimal control problems

Recently published results of Gift (Ref. 1) are concerned with the necessary conditions for singular optimal control problems (in the sense of Pontryagin's minimum principle). However, those results are incorrect. An illustrative counterexample is given here.This paper was written while the author studied at Fudan University. The author wishes to thank Professor X. J. Li for his guidance and encouragement. The research was partially supported by NSF of China and the Chinese State Education Commission NSF.     

Stability and sensitivity of solutions to nonlinear optimal control problems

Parameter-dependent optimal control problems for nonlinear ordinary differential equations, subject to control and state constraints, are considered. Sufficient conditions are formulated under which the solutions and the associated Lagrange multipliers are locally Lipschitz continuous and directionally differentiable functions of the parameter. The directional derivatives are characterized.This research was partially supported by Grant No. 3 0256 91 01 from Komitet Bada Naukowych.     

Multiplicity results for superlinear boundary value problems with impulsive effects

We study the existence and multiplicity of solutions for a nonlinear boundary value problem subject to perturbations of impulsive terms. Under suitable assumptions on the potential of the nonlinearity, the existence of one or two solutions is established. Our approach is based on a local minimum theorem and a two non‐zero critical points for differentiable functionals. Copyright © 2015 John Wiley & Sons, Ltd.     

On some elliptic optimal control problems with state constraints

In this paper optimality conditions will be derived for elliptic optimal control problems with a restriction on the state or on the gradient of the state. Essential tools are the method of transposition and generalized trace theorems and green's formulas from the theory of elliptic differential equations.     

A new approach to the Pontryagin maximum principle for nonlinear fractional optimal control problems

In this paper, we discuss a new general formulation of fractional optimal control problems whose performance index is in the fractional integral form and the dynamics are given by a set of fractional differential equations in the Caputo sense. The approach we use to prove necessary conditions of optimality in the form of Pontryagin maximum principle for fractional nonlinear optimal control problems is new in this context. Moreover, a new method based on a generalization of the Mittag–Leffler function is used to solving this class of fractional optimal control problems. A simple example is provided to illustrate the effectiveness of our main result. Copyright © 2016 John Wiley & Sons, Ltd.     

Stabilization of neutral functional differential equations

In this paper, we prove a necessary and sufficient condition for feedback stabilization of neutral functional differential equations.This paper follows the program of the Gruppo Nazionale per l'Analisi Funzionale e le Sue Applicazioni, Consiglio Nazionale delle Ricerche (CNR), Rome, Italy.     

The Lagrange-Newton method for nonlinear optimal control problems

We investigate local convergence of the Lagrange-Newton method for nonlinear optimal control problems subject to control constraints including the situation where the terminal state is fixed. Sufficient conditions for local quadratic convergence of the method based on stability results for the solutions of nonlinear control problems are discussed.     

An optimal control problem for systems described by partial differential equations of hyperbolic type with delay

An optimal control problem of the Gourse type with delay is investigated. With a given aim functional, a necessary condition of optimality is formulated and proved in the form of a maximum principle. The proof is based on the reduction of a problem with delay to a problem without delay.The authors thank Prof. G. Leitmann, University of California, Berkeley, for discussions and for his interest in this paper.     

On reachable set of singularly perturbed differential inclusions and optimal control problems

Asymptotic properties of reachable sets of singularly perturbed differential inclusions are given. Their applications to optimal control problems are also studied     

A maximum principle for optimal control problems with mixed constraints

Necessary conditions in the form of maximum principles are derivedfor optimal control problems with mixed control and state constraints.Traditionally, necessary condtions for problems with mixed constraintshave been proved under hypothesis which include the requirementthat the Jacobian of the mixed constraint functional, with respectto the control variable, have full rank. We show that it canbe replaced by a weaker ‘interiority’ hypothesis.This refinement broadens the scope of the optimality conditions,to cover some optimal control problems involving differentialalgebraic constraints, with index greater than unity.     

An elementary proof of the maximum principle for optimal control problems governed by a Volterra integral equation

An elementary proof of the maximum principle for optimal control problems whose states are governed by Volterra integral equations is given. Our proof is motivated by the work of Michel (Ref. 7) and utilizes only elementary results from analysis and mathematical programming. By appealing to Pontryagin-type perturbations of the controls, the above optimal control problem is effectively reduced to a mathematical programming problem. The results are then obtained by appealing to well-known mathematical programming results.     

Some results,concerning the variation of eigenvalues,when these depend on a real parameter

In this article we deal with a Hamiltonial of the form H(v) = Ho + A(v) where Ho is a self-adjoint bounded or unbounded operator on a Hilbert space and A(v) is a bounded self-adjoint perturbation depending on a real parameter v. In quantum mechanics a variety of results has been obtained by taking formally the derivative of the eigenvectors and eigenvalues of H(v).The differentiability of the eigenvectors and eigenvalues has been rigorously proved under several assumptions. Among these assumptions is the assumption that the eigenvalues are simple and the assumption that the perturbation A(v) is a uniformly bounded self-adjoint operator. A part of this article is dealing with examples, which show that these two assumptions are essential. The rest of this article is devoted to different applications concerning asymptotic relations of eigenvalues and a result for the solutions of the equation d_y/d_t= M(t)y in an abstract infinite dimensional Hilbert space, where iM(t)(1²=-1) is self-adjoint for every t in an interval. This result finds a succesful application to the theory of Toda and Langmuir lattices.     

Some new results in differential algebraic control theory

In the differential algebra framework, static state and dynamicstate feedback linearization are considered. The relationshipbetween dynamic feedback and flatness defined by Fliess is discussed.The existence of an equivalent proper differential I–Osystem of a given differential I–O system is discussed,which is closely related to the choice of a proper fictitiousoutput in control design. The concept flatness and its relationwith dynamic feedback linearizability, controllability, observability,invertibility and minimal realization are discussed. Finally,it is demonstrated that many fundamental control concepts andtheir interrelationships can be incorporated into an extendedcontrol diagram.     

Existence of an optimal control for stochastic systems governed by ito equations

We give existence theorems for stochastic control problems with a lower semicontinuous cost functional and governed by Ito equations. We prove that two formulations of the fundamental problem are equivalent, one involving nonanticipative controls and the other involving (measurable) feedback controls. We then use the concept ofconvergence in distribution to prove existence for the first problem, and hence for the second as well. While our work has certain similarities with a paper of Kushner, our techniques are different and lead to more general results.