Stochastic maximum principle for mean-field forward-backward stochastic control system with terminal state constraints

In this paper,we consider an optimal control problem with state constraints,where the control system is described by a mean-field forward-backward stochastic differential equation(MFFBSDE,for short)and the admissible control is mean-field type.Making full use of the backward stochastic differential equation theory,we transform the original control system into an equivalent backward form,i.e.,the equations in the control system are all backward.In addition,Ekeland’s variational principle helps us deal with the state constraints so that we get a stochastic maximum principle which characterizes the necessary condition of the optimal control.We also study a stochastic linear quadratic control problem with state constraints.     

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.     

First- and second-order optimality conditions for a strong local minimum in control problems with pure state constraints

In this paper first- and second-order optimality conditions for a strong local minimum are presented for optimal control problems with pure state set-inclusion constraints. The first-order condition is of Pontryagin type, while the second-order condition is of the form of an accessory problem associated with the strong local minimality. This latter condition contains an extra term reflecting the presence of the pure state constraints.     

Sufficient second-order optimality conditions for convex control constraints

In this article sufficient optimality conditions are established for optimal control problems with pointwise convex control constraints. Here, the control is a function with values in Rn. The constraint is of the form u(x)∈U(x), where U is a set-valued mapping that is assumed to be measurable with convex and closed images. The second-order condition requires coercivity of the Lagrange function on a suitable subspace, which excludes strongly active constraints, together with first-order necessary conditions. It ensures local optimality of a reference function in an L^∞-neighborhood. The analysis is done for a model problem namely the optimal distributed control of the instationary Navier-Stokes equations.     

Nonlinear and chaos control of a micro-electro-mechanical system by using second-order fast terminal sliding mode control

In this paper, a novel second-order fast terminal sliding mode control (SFTSMC) scheme is proposed to suppress the chaotic motion of a micro-mechanical resonator with system uncertainty and external disturbance. To obtain a better disturbance rejection property, a fuzzy logic system is introduced to estimate the upper boundary of the sum of system uncertainty and external disturbance. Moreover, we employ the finite-time technique to obtain the properties of fast response and high precision. Finally, numerical simulations demonstrate the effectiveness of the proposed control scheme.     

Sufficient conditions for optimal control with state and control constraints

A monotonicity result is utilized to derive sufficient optimality conditions of considerable generality for an individual trajectory in control theory. The sufficiency theorem embodying these conditions generalizes those of Boltyanskii and Leitmann and is applied to a simple control system to which their sufficiency theorems are not applicable. Conditions on the state equations and state space are completely relaxed. The set of admissible controls is extended to the set of measurable controls and the integrand of the performance index has its membership extended to the class of bounded Borel-measurable functions. The decomposition of the state space is required to be onlyplain denumerable.     

No-gap second-order optimality conditions for optimal control problems with a single state constraint and control

The paper deals with optimal control problems with only one control variable and one state constraint, of arbitrary order. We consider the case of finitely many boundary arcs and touch times. We obtain a no-gap theory of second-order conditions, allowing to characterize second-order quadratic growth.     

Necessary second-order conditions for a weak local minimum in a problem with endpoint and control constraints

One of the first steps towards necessary second-order optimality conditions in problems with constraints was taken by Dubovitskii and Milyutin in 1965. They offered a scheme that was very effective in smooth optimization problems, but seemed to be not suitable for applications in problems with pointwise control constraints. In this article we consider a modification of the Dubovitskii–Milyutin scheme, which allows to derive necessary second-order conditions for a weak local minimum in optimal control problems with a finite number of endpoint constraints of equality and inequality type and with pointwise control constraints of inequality type given by smooth functions. Assuming that the gradients of active control constraints are linearly independent, we provide rather straightforward proof of these conditions for a measurable and essentially bounded optimal control.     

On a control problem for a nonlinear second-order system

A control problem for a nonlinear second-order system of differential equations in the presence of uncontrollable effects is investigated. A solution algorithm is proposed in the case when one phase coordinate of the system is measured at discrete moments. The algorithm is stable with respect to information noises and computational errors. Results of a computer experiment are presented.     

Necessary conditions for optimal control problems with state constraints

Necessary conditions of optimality are derived for optimal control problems with pathwise state constraints, in which the dynamic constraint is modelled as a differential inclusion. The novel feature of the conditions is the unrestrictive nature of the hypotheses under which these conditions are shown to be valid. An Euler Lagrange type condition is obtained for problems where the multifunction associated with the dynamic constraint has values possibly unbounded, nonconvex sets and satisfies a mild `one-sided' Lipschitz continuity hypothesis. We recover as a special case the sharpest available necessary conditions for state constraint free problems proved in a recent paper by Ioffe. For problems where the multifunction is convex valued it is shown that the necessary conditions are still valid when the one-sided Lipschitz hypothesis is replaced by a milder, local hypothesis. A recent `dualization' theorem permits us to infer a strengthened form of the Hamiltonian inclusion from the Euler Lagrange condition. The necessary conditions for state constrained problems with convex valued multifunctions are derived under hypotheses on the dynamics which are significantly weaker than those invoked by Loewen and Rockafellar to achieve related necessary conditions for state constrained problems, and improve on available results in certain respects even when specialized to the state constraint free case.

Proofs make use of recent `decoupling' ideas of the authors, which reduce the optimization problem to one to which Pontryagin's maximum principle is applicable, and a refined penalization technique to deal with the dynamic constraint.

     

Optimal control computation for linear time-lag systems with linear terminal constraints

A computational scheme using the technique of control parameterization is developed for solving a class of optimal control problems involving linear hereditary systems with bounded control region and linear terminal constraints. Several examples have been solved to illustrate the efficiency of the technique.The authors wish to thank Dr. B. D. Craven for pointing out an error in an earlier version of this paper.From January 1985, Associate Professor, Department of Industrial and Systems Engineering, National University of Singapore, Kent Ridge, Singapore.     

On optimal control problems with state constraints

This paper is concerned with necessary conditions for a general optimal control problem developed by Russak and Tan. It is shown that, in most cases, a further relation between the multipliers holds. This result is of interest in particular for the investigation of perturbations of the state constraint.     

Boundary optimal flow control with state constraints

The numerical solution of the Dirichlet boundary optimal control problem of the Navier-Stokes equations in presence of pointwise state constraints is investigated. A Moreau-Yosida regularization of the problem is proposed to obtain regular multipliers. Optimality conditions are derived and the convergence of the regularized solutions towards the original one is presented. The paper ends with a numerical experiment. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On infinite-dimensional control systems with state and control constraints

In this paper we examine infinite-dimensional control systems governed by semilinear evolution equations and having both state and control constraint. We introduce the relaxed system and show that the original trajectories are dense in an appropriate function space in the relaxed ones. We also determine the dependence of the solution set on the initial conditions. Then using those results we establish necessary and sufficient conditions for optimality for some optimization problems. Finally we prove some controllability results.     

Computational algorithm for a distributed optimal control problem of parabolic type with terminal inequality constraints

A corrigendum to Ref. 1 is given.     

Computational algorithm for a distributed optimal control problem of parabolic type with terminal inequality constraints

A computational algorithm for optimal control problems with control and terminal inequality constraints involving first boundary-value problems of parabolic type is presented. The convergence properties are also studied.This work, which was partly supported by the Australian Research Grants Committee, was done during the period when Z. S. Wu was an Honorary Visiting Fellow in the School of Mathematics at the University of New South Wales, Australia.     

Error estimates for parabolic optimal control problems with control and state constraints

The numerical approximation to a parabolic control problem with control and state constraints is studied in this paper. We use standard piecewise linear and continuous finite elements for the space discretization of the state, while the dG(0) method is used for time discretization. A priori error estimates for control and state are obtained by an improved maximum error estimate for the corresponding discretized state equation. Numerical experiments are provided which support our theoretical results.     

A-posteriori error estimates for optimal control problems with state and control constraints

We discuss the full discretization of an elliptic optimal control problem with pointwise control and state constraints. We provide the first reliable a-posteriori error estimator that contains only computable quantities for this class of problems. Moreover, we show, that the error estimator converges to zero if one has convergence of the discrete solutions to the solution of the original problem. The theory is illustrated by numerical tests.     

Approximation of optimal control problems with state constraints

We study the approximation of control problems governed by elliptic partial differential equations with pointwise state constraints. For a finite dimensional approximation of the control set and for suitable perturbations of the state constraints, we prove that the corresponding sequence of discrete control problems converges to a relaxed problem. A similar analysis is carried out for problems in which the state equation is discretized by a finite element method.     

Sufficient optimality conditions for nonconvex control problems with state constraints

The sufficient optimality conditions of Zeidan for optimal control problems (Refs. 1 and 2) are generalized such that they are applicable to problems with pure state-variable inequality constraints. We derive conditions which neither assume the concavity of the Hamiltonian nor the quasiconcavity of the constraints. Global as well as local optimality conditions are presented.