Game problems for fractional-order linear systems

The paper is concerned with studying approach game problems for linear conflict-controlled processes with fractional derivatives of arbitrary order. Namely, the classical Riemann-Liouville fractional derivatives, Dzhrbashyan-Nersesyan or Caputo regularized derivatives, and Miller-Ross sequential derivatives are considered. Under fixed controls of the players, solutions are presented in the form of analogs of the Cauchy formula with the use of generalized matrix Mittag-Leffler functions. The investigation is based on the method of resolving functions, which allows one to obtain sufficient conditions for the termination of the approach problem in some guaranteed time period. The results are exemplified by model game problems with a simple matrix and separated motions of fractional order π and e.     

Optimal control of linear stochastic systems described by functional differential equations

The optimal control is determined for a class of systems by assuming the configuration of the feedback loop. The feedback loop consists of an unbiased estimator and controller. The gain matrices of the estimator and the controller are so determined that the mean-squared estimation error and the average value of a quadratic cost functional, respectively, are minimized. This is accomplished by the application of the matrix maximum principle to a distributed parameter system. The results indicate that the optimal estimation and the optimal control can be computed independently (separation principle).This work was supported in part by the Air Force Office of Scientific Research, Grant No. 69-1776.     

Optimal control problems for wave equations

The problem on minimizing a quadratic functional on trajectories of the wave equation is considered. We assume that the density of external forces is a control function. A control problem for a partial differential equation is reduced to a control problem for a countable system of ordinary differential equations by use of the Fourier method. The controllability problem for this countable system is considered. Conditions of the noncontrollability for some wave equations were obtained.     

Optimal control for systems described by hyperbolic equations with strong nonlinearity

An optimization control problem for a hyperbolic equation is considered. The system is nonlinear with respect to the state derivative. The regularization technique for the state equation is applied. The necessary conditions of optimality for the regularized control problem are proved. It uses the extended differentiability of the control-state mapping for the regularized equation. The convergence of the regularization method is proved. Thus the optimal control for the regularized problem with a small enough regularization parameter can be chosen as an approximate solution of the initial optimization problem.     

Optimal control problems with state constraint governed by Navier-Stokes equations

This work deals with the existence of optimal solution and the maximum principle for optimal control problem governed by Navier-Stokes equations with state constraint of pointwise type in 3D. Strong results in 2D are also given.     

Oblique derivative problems for linear mixed equations of second order

Oblique derivative boundary value problems for the linear mixed (elliptic-hyperbolic) equation of the second order, i.e. the generalized Lavrent’ev-Bitsadze equation with weak conditions are discussed. The representation of solutions for the above boundary value problem is given, the uniqueness and existence of solutions of the above problem are proved, and a priori estimates of the solutions of the above problem are obtained. Project supported by the National Natural Science Foundation of China.     

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.     

Numerical approximation of nonconvex optimal control problems defined by parabolic equations

In this paper, we consider an optimal control problem for distributed systems governed by parabolic equations. The state equations are nonlinear in the control variable; the constraints and the cost functional are generally nonconvex. Relaxed controls are used to prove existence and derive necessary conditions for optimality. To compute optimal controls, a descent method is applied to the resulting relaxed problem. A numerical method is also given for approximating a special class of relaxed controls, notably those obtained by the descent method. Convergence proofs are given for both methods, and a numerical example is provided.     

Optimal control problems governed by some semilinear parabolic equations

This paper is concerned with the optimal control problem for the Keller–Segel equations. That is, as a generalization of Ryu and Yagi (J. Math. Anal. Appl. 256 (2001) 45–66) we derive the optimality conditions for the optimal control problem governed by a semilinear abstract equation of nonmonotone type. Moreover, we prove the uniqueness of the optimal control under some conditions.     

Boundary value problems for systems of linear evolution equations

It is shown that the new method for solving initial-boundaryvalue problems for scalar evolution equations recently introducedby one of the authors can also be applied to systems of evolutionequations. The novel step needed in this case is the constructionof a scalar Lax pair by using a suitable parametrization ofthe dispersion relation as well as certain linear transformations.The simultaneous spectral analysis of the Lax pair yields thesolution of a given initial-boundary value problem in termsof an integral in the complex spectral plane which involvesan appropriate x-transform of the initial conditions and anappropriate t-transform of the boundary conditions. These transformsare neither the x-Fourier transform nor the t-Laplace transform,rather they are new transforms custom made for the given systemof partial differential equations (PDEs) and the given domain.This method is illustrated by solving on the half-line the linearizedequations governing infinitesimal deformations in a heat conductingbar. Received 8 January 2003. Revised 18 July 2003. ^* Email: p.a.treharne{at}damtp.cam.ac.uk Email: t.fokas{at}damtp.cam.ac.uk     

Optimal control of systems with discontinuous differential equations

In this paper we discuss the problem of verifying and computing optimal controls of systems whose dynamics is governed by differential systems with a discontinuous right-hand side. In our work, we are motivated by optimal control of mechanical systems with Coulomb friction, which exhibit such a right-hand side. Notwithstanding the impressive development of nonsmooth and set-valued analysis, these systems have not been closely studied either computationally or analytically. We show that even when the solution crosses and does not stay on the discontinuity, differentiating the results of a simulation gives gradients that have errors of a size independent of the stepsize. This means that the strategy of “optimize the discretization” will usually fail for problems of this kind. We approximate the discontinuous right-hand side for the differential equations or inclusions by a smooth right-hand side. For these smoothed approximations, we show that the resulting gradients approach the true gradients provided that the start and end points of the trajectory do not lie on the discontinuity and that Euler’s method is used where the step size is “sufficiently small” in comparison with the smoothing parameter. Numerical results are presented for a crude model of car racing that involves Coulomb friction and slip showing that this approach is practical and can handle problems of moderate complexity.     

Optimal control problems for semilinear non-well-posed parabolic differential equations

This paper deals with necessary conditions for optimal control problem governed by some semilinear parabolic differential equation which may be non-well-posed. State constrained problem is considered. Finally, under some suitable assumptions, we obtain the existence of optimal pairs.     

Optimal control problems with state constraints for semilinear distributed-parameter systems

We consider optimal control problems for distributed-parameter systems described by semilinear equations, with constraints on the control and on the state, and an exact pointwise target condition. As an application of a general theory of nonlinear programming problems in Banach spaces, a version of the Pontryagin maximum principle is obtained.This research was partly supported by the National Science Foundation under Grant DMS-92-21819.     

Optimal projection equations for reduced-order modelling,estimation, and control of linear systems with multiplicative white noise

The optimal projection equations for quadratically optimal reduced-order modelling, estimation, and control are generalized to include the effects of state, control, and measurement dependent noise.     

Optimal control with impulsive component for systems described by implicit parabolic operator differential equations

We study the problem of optimal control with impulsive component for systems described by abstract Sobolev-type differential equations with unbounded operator coefficients in Hilbert spaces. The operator coefficient of the time derivative may be noninvertible. The main assumption is a restriction imposed on the resolvent of the characteristic operator pencil in a certain right half plane. Applications to Sobolevtype partial differential equations are discussed.     

Nonlocal boundary-value problems for systems on linear partial differential equations

We study the classical well-posedness of problems with nonlocal two-point conditions for typeless systems of linear partial differential equations with variable coefficients in a cylindrical domain. We prove metric theorems on lower bounds for small denominators that appear in the construction of solutions of such problems.