On an optimal control problem

In this paper, we applied the finite differences method to the solution of variational problem of an inverse problem for the Scrödinger equation with a final functional. These types of problems arise in various fields in quantum-mechanical, nuclear physics and modern physics [2, 11]. Also, we prove two estimates for the differences scheme and convergence speed of difference approximations according to the functional. The inverse problems for the Schrödinger equation having different variational formulation were investigated in [7, 12, 13].     

Local convergence of an algorithm for solving optimal control problems

In this paper, a method is proposed for the numerical solution of optimal control problems with terminal equality constraints. The multiplier method is employed to deal with the terminal equality constraints. It is shown that a sequence of control functions, which converges to the optimal control, is obtained by the alternate update of control functions and multipliers.The authors wish to thank Dr. N. Fujii for his most valuable comments and suggestions.     

On sensitivity in an optimal control problem

We give a new criterion for the existence of value in differential games. The method of proof involves Lipschitz differential games and hence extends to games with more general dynamics. The connection between using measurable control functions or simply constants is clarified.     

Convergence properties of an algorithm for solving non-differentiable optimal control problems

An algorithm for solving the open-loop optimal control problem proposed by Y. Sakawa and Y. Shindo is analysed. The algorithm is essentially based on the use of the Pontriagin maximum principle. The convergence properties of the algorithm are investigated. It is established here, under weaker assumptions, that i) the algorithm is well-defined, ii) the cost functional monotonically decreases, iii) a weak necessary optimality condition is asymptotically verified, if the sequence of computational control parameters is bounded from above and from below.     

On an optimal control problem with an integral functional of a rational control function

We present the solution of an optimization problem with integral performance functional that is nonlinear with respect to the control and contains a discounting parameter in the class of programmed controls under two-sided control constraints. The optimal control is found in the form of a function of time (a program). On the basis of the theoretical results, we perform numerical experiments with model and real data.     

On optimal control by coefficients in an elliptic equation

We consider a problem of optimal control by coefficients in a linear elliptic equation. We study the well-posedness of the statement of the problem and obtain a necessary optimality condition.     

Sequential gradient-restoration algorithm for optimal control problems

This paper considers the problem of minimizing a functionalI which depends on the statex(t), the controlu(t), and the parameter . Here,I is a scalar,x ann-vector,u anm-vector, and ap-vector. At the initial point, the state is prescribed. At the final point, the statex and the parameter are required to satisfyq scalar relations. Along the interval of integration, the state, the control, and the parameter are required to satisfyn scalar differential equations. Asequential algorithm composed of the alternate succession of gradient phases and restoration phases is presented. This sequential algorithm is contructed in such a way that the differential equations and boundary conditions are satisfied at the end of each iteration, that is, at the end of a complete gradient-restoration phase; hence, the value of the functional at the end of one iteration is comparable with the value of the functional at the end of any other iteration.In thegradient phase, nominal functionsx(t),u(t), satisfying all the differential equations and boundary conditions are assumed. Variations x(t), u(t), leading to varied functions (t),(t), are determined so that the value of the functional is decreased. These variations are obtained by minimizing the first-order change of the functional subject to the linearized differential equations, the linearized boundary conditions, and a quadratic constraint on the variations of the control and the parameter.Since the constraints are satisfied only to first order during the gradient phase, the functions (t),(t), may violate the differential equations and/or the boundary conditions. This being the case, a restoration phase is needed prior to starting the next gradient phase. In thisrestoration phase, the functions (t),(t), are assumed to be the nominal functions. Variations (t), (t), leading to varied functions (t),û(t), consistent with all the differential equations and boundary conditions are determined. These variations are obtained by requiring the least-square change of the control and the parameter subject to the linearized differential equations and the linearized boundary conditions. Of course, the restoration phase must be performed iteratively until the cumulative error in the differential equations and boundary conditions becomes smaller than some preselected value.If the gradient stepsize is , an order-of-magnitude analysis shows that the gradient corrections are x=O(), u=O(), =O(), while the restoration corrections are . Hence, for sufficiently small, the restoration phase preserves the descent property of the gradient phase: the functionalI decreases between any two successive iterations.Methods to determine the gradient stepsize in an optimal fashion are discussed. Examples are presented for both the fixed-final-time case and the free-final-time case. The numerical results show the rapid convergence characteristics of the sequential gradient-restoration algorithm.The portions of this paper dealing with the fixed-final-time case were presented by the senior author at the 2nd Hawaii International Conference on System Sciences, Honolulu, Hawaii, 1969. The portions of this paper dealing with the free-final-time case were presented by the senior author at the 20th International Astronautical Congress, Mar del Plata, Argentina, 1969. This research, supported by the NASA-Manned Spacecraft Center, Grant No. NGR-44-006-089, Supplement No. 1, is a condensation of the investigations presented in Refs. 1–5. The authors are indebted to Professor H. Y. Huang for helpful discussions.     

On an optimal control problem for weakly nonlinear hyperbolic equations

The correctness of the problem of optimal control by the coefficients for a weakly nonlinear hyperbolic equation is studied, and a necessary condition of optimality is established.     

On the existence of an optimal feedback control for stochastic systems

We prove the existence of an optimal control for systems of stochastic differential equations without solving the Bellman dynamic programming equation. Instead, we use direct methods for solving extremal problems.     

On an optimal control problem for a hyperbolic partial differential equation

Existence of solutions is proved for a minimum problem for a distributed-parameter control system described by a linear, hyperbolic partial differential equation. The cost function is an integral depending on boundary controls.This research was supported by the Consiglio Nazionale delle Ricerche, Rome, Italy.     

On an optimal boundary control problem with a dynamic boundary condition

We are the first solve the optimization problem for a boundary displacement control of string vibrations at one endpoint in the case of a nonstationary boundary condition containing a directional derivative at the other endpoint.     

On an optimal control problem for a distributed parameter control process with unbounded coefficients

Let $$\begin{gathered} z_{xy} + A\left( {x,y} \right)z_x + B\left( {x,y} \right)z_y + C\left( {x,y} \right)z = U\left( {x,y} \right), \hfill \\ a.e. in \Delta = ]0,a[ \times ]0,b[, \hfill \\ \end{gathered}$$ be a distributed parameter control process, where the target consists of the boundary data on the two sides of \(\bar \Delta\) ( \(\bar \Delta\) is the closure of Δ), which do not contain the origin. For such a process, Pulvirenti and Santagati (Refs. 1–5) have investigated various control and optimal control problems, subject to the following assumption for the coefficients:A,B,C,A _x ,B _y ∈C°( \(\bar \Delta\) ). In this paper, the above process is considered under more general assumptions on the coefficients, which now need not be bounded functions. A minimization problem, where the cost is an integral depending on the controls (which are the functionU and the boundary data on the two sides of \(\bar \Delta\) which contain the origin), is studied. Some existence theorems are established. To this aim, we first prove existence, uniqueness, and continuous dependence on the controls of the response of the process.     

On the existence of an optimal element for a delay control problem

The existence theorems of the optimal element are proved for a nonlinear control problem with constant delay in phase coordinates and with general functional. Here element implies the collection of delay parameter and initial function, initial moment and vector, control and finally moment.     

On the optimal control of stochastic systems with an exponential-of-integral performance index

In this paper a theory of optimal control is developed for stochastic systems whose performance is measured by the exponential of an integral form. Such a formulation of the cost function is shown to be not only general and useful but also analytically tractable. Starting with very general classes of stochastic systems, optimality conditions are obtained which exploit the multiplicative decomposability of the exponential-of-integral form. Specializing to partially observed systems of stochastic differential equations with Brownian Motion disturbances, optimality conditions are obtained which parallel those for systems with integral costs. Also treated are the special cases of linear systems with exponential of quadratic costs for which explicit optimal controls are obtainable. In addition, several general results of independent interest are obtained, which concern optimality of stochastic systems.     

A numerical algorithm for singular optimal LQ control systems

A numerical algorithm to obtain the consistent conditions satisfied by singular arcs for singular linear–quadratic optimal control problems is presented. The algorithm is based on the Presymplectic Constraint Algorithm (PCA) by Gotay-Nester (Gotay et al., J Math Phys 19:2388–2399, 1978; Volckaert and Aeyels 1999) that allows to solve presymplectic Hamiltonian systems and that provides a geometrical framework to the Dirac-Bergmann theory of constraints for singular Lagrangian systems (Dirac, Can J Math 2:129–148, 1950). The numerical implementation of the algorithm is based on the singular value decomposition that, on each step, allows to construct a semi-explicit system. Several examples and experiments are discussed, among them a family of arbitrary large singular LQ systems with index 2 and a family of examples of arbitrary large index, all of them exhibiting stable behaviour. Research partially supported by MEC grant MTM2004-07090-C03-03. SIMUMAT-CM, UC3M-MTM-05-028 and CCG06-UC3M/ESP-0850.     

On sensitivity in optimal control problems

Two types of interpretations of multipliers in both static and dynamic optimization problems are described. It is snown that the Lagrange multipliers encountered in mathematical programming problems and the auxiliary functions arising in Pontryagintype optimal control problems sometimes have highly analogous interpretations as rates of change of the optimal attainable value of an objective function, or in some cases as bounds on average rates of change.     

On a student-related optimal control problem

In a recent paper (Ref. 1), Cheng and Teo discussed some further extensions of a student-related optimal control problem which was originally proposed by Raggettet al. (Ref. 2) and later on modified by Parlar (Ref. 3). In this paper, we treat further extensions of the problem.This paper is a modified and improved version of Ref. 4. It is based, in part, on research sponsored by NSF.     

On optimal control over quasilinear systems

We consider the problem of approximately optimal stabilization of quasilinear systems with geometric constraints imposed on control. By using the idea of Krotov global estimates, we justify a method for approximation of the optimal stabilization control and estimate an error in terms of a functional.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1280–1294, September, 1995.     

Minimax algorithm for constructing an optimal control strategy in differential games with a Lipschitz payoff

For a zero-sum differential game, an algorithm is proposed for computing the value of the game and constructing optimal control strategies with the help of stepwise minimax. It is assumed that the dynamics can be nonlinear and the cost functional of the game is the sum of an integral term and a terminal payoff function that satisfies the Lipschitz condition but can be neither convex nor concave. The players’ controls are chosen from given sets that are generally time-dependent and unbounded. An error estimate for the algorithm is obtained depending on the number of partition points in the time interval and on the fineness of the spatial triangulation. Numerical results for an illustrative example are presented.     

Remarks on the algorithm of sakawa for optimal control problems

A general optimal control problem for ordinary differential equations is considered. For this problem, some improvements of the algorithm of Sakawa are discussed. We avoid any convexity assumption and show with an example that the algorithm is even applicable in cases in which no optimal control exists.