Generalized solutions for singular optimal control problems

This text presents a complete theory of existence/uniqueness and the structure of generalized solutions for singular linear-quadratic optimal control problems. Generalized optimal controls are distributions of order r and the corresponding generalized trajectories are distributions of order (r − 1). r is the “order of singularity” of the problem, an integer no greater than the dimension of the state space. Its value is obtained through a certain reduction procedure. In the final section, some perspectives and partial results concerning the extension of these results to nonlinear problems are briefly discussed. Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 27, Optimal Control, 2007.     

Function-space quasi-Newton algorithms for optimal control problems with bounded controls and singular arcs

Two existing function-space quasi-Newton algorithms, the Davidon algorithm and the projected gradient algorithm, are modified so that they may handle directly control-variable inequality constraints. A third quasi-Newton-type algorithm, developed by Broyden, is extended to optimal control problems. The Broyden algorithm is further modified so that it may handle directly control-variable inequality constraints. From a computational viewpoint, dyadic operator implementation of quasi-Newton methods is shown to be superior to the integral kernel representation. The quasi-Newton methods, along with the steepest descent method and two conjugate gradient algorithms, are simulated on three relatively simple (yet representative) bounded control problems, two of which possess singular subarcs. Overall, the Broyden algorithm was found to be superior. The most notable result of the simulations was the clear superiority of the Broyden and Davidon algorithms in producing a sharp singular control subarc.This research was supported by the National Science Foundation under Grant Nos. GK-30115 and ENG 74-21618 and by the National Aeronautics and Space Administration under Contract No. NAS 9-12872.     

A geometric computing method for nonlinear optimal regulator problems with singular arcs

A computing algorithm, based on the geometry of certain reachable sets, is presented for fixed terminal time optimal regular problems having differential equations \(\dot x = f(x ,u , t)\) . Admissible controls must be measurable and have values in a setU, which must be compact, but need not be convex. Functionsf(x, u, t) andf _x (x, u, t) must be continuous and Lipschitz inx andu, but existence off _u (x, u, t) or second derivatives is not required. The algorithm is based on taking a sequence of nonlinear steps, each of which linearizes \(\dot x = f(x ,u , t)\) in state only, about a current nominal control and trajectory. Small perturbations are assured by keeping the perturbed controlclose to the nominal control. In each nonlinear step, a regulator problem,linear in state, is solved by a convexity method of Barr and Gilbert (Refs. 1–2), which is undeterred by the possibility of singular arcs. The resulting control function is substituted into the original nonlinear differential equations, producing an improved trajectory. Convergence of the algorithm is not proved, but demonstrated by a computing example, known to be singular. In addition, procedures are described for choosing parameters in the algorithm and for testing for theplausibility of convergence.     

On the order of singular optimal control problems

In singular optimal control problems, the functional form of the optimal control function is usually determined by solving the algebraic equation which results by successively differentiating the switching function until the control appears explicitly. This process defines the order of the singular problem. Order-related results are developed for singular linear-quadratic problems and for a bilinear example which gives new insights into the relationship between singular problem order and singular are order.Dedicated to R. BellmanThis work was supported by the National Science Foundation under Grant No. ENG-77-16660.     

Ordered optimal solutions and parametric minimum cut problems

In this paper, we present an algebraic sufficient condition for the existence of a selection of optimal solutions in a parametric optimization problem that are totally ordered, but not necessarily monotone. Based on this result, we present necessary and sufficient conditions that ensure the existence of totally ordered selections of minimum cuts for some classes of parametric maximum flow problems. These classes subsume the class studied by Arai et al. [Discrete Appl. Math. 41 (1993) 69–74] as a special case.     

Asymptotic properties of singular solutions in degenerate parabolic equation with boundary flux

ABSTRACT

This paper deals with blow-up and quenching solutions of degenerate parabolic problem involving m-Laplacian operator and nonlinear boundary flux. The blow-up and quenching criteria are classified under the conditions on the initial data but with less conditions on the relationship among the exponents, respectively. Moreover, asymptotic properties including singular rates, set and time estimates are determined for the blow-up solutions and the quenching solutions, respectively.     

Second-order optimality principle for singular optimal control problems

A new necessary condition for singular optimal control problems is presented in this paper. The condition is simpler to apply than existing conditions and is easily derived from a Taylor series expansion of the performance index.     

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.     

Necessary conditions for optimal singular stochastic control problems

In this paper, necessary conditions of optimality, in the form of a maximum principle, are obtained for singular stochastic control problems. This maximum principle is derived for a state process satisfying a general stochastic differential equation where the coefficient associated to the control process can be dependent on the state, extending earlier results of the literature.     

Some properties of solutions to the singular quasilinear problems

In this paper, a kind of quasilinear elliptic problem is studied, which involves the critical exponent and singular potentials. By the Caffarelli-Kohn-Nirenberg inequality and variational methods, some important properties of the positive solution to the problem are established.     

Asymptotic solution of singularly perturbed linear-quadratic optimal control problems with discontinuous coefficients

An asymptotic solution of a singularly perturbed linear-quadratic optimal control problem with discontinuous coefficients is constructed by directly substituting an boundary-layer asymptotic expansion of the solution into the condition of the problem and considering a series of problems for finding the asymptotic terms. The error in the approximate solution is estimated. It is shown that the values of the minimized functional do not increase when the next approximations of the optimal control are used.     

Existence of finitely optimal solutions for infinite-horizon optimal control problems

In this paper, we investigate the existence of finitely optimal solutions for the Lagrange problem of optimal control defined on [0, ) under weaker convexity and seminormality hypotheses than those of previous authors. The notion of finite optimality has been introduced into the literature as the weakest of a hierarchy of types of optimality that have been defined to permit the study of Lagrange problems, arising in mathematical economics, whose cost functions either diverge or are not bounded below. Our method of proof requires us to analyze the continuous dependence of finite-interval Lagrange problems with respect to a prescribed terminal condition. Once this is done, we show that a finitely optimal solution can be obtained as the limit of a sequence of solutions to a sequence of corresponding finite-horizon optimal control problems. Our results utilize the convexity and seminormality hypotheses which are now classical in the existence theory of optimal control.This research forms part of the author's doctoral dissertation written at the University of Delaware, Newark, Delaware under the supervision of Professor Thomas S. Angell.     

The asymptotics of constrained control in optimal singular parabolic problems

A complete asymptotic solution is constructed and justified for the optimal singular parabolic problems with constrained control and a completely degenerate differential part of an operator.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 10, pp. 1345–1355, October, 1993.     

Asymptotic properties of solutions to some three-dimensional wave problems

Green's function for the Helmholtz equation of a bounded domain D with Neumann boundary conditions is considered. The boundary of D is smooth and bounded, and its Gaussian curvature is positive. An estimate of Green's function is obtained in the nonphysical region 0 >.Jm K >–|Re K |^1/3.This estimate shows that the nonspectral singularities of Green's function lie below the parabola Jm K0 ; |Jm K|=|Re K|^1/3.On the basis of these results, the authors investigate the behavior as t of Green's function for the wave equation in the domain Dx(0相似文献   

Asymptotic analysis of a bounded control in optimal elliptic problems

An asymptotics of a bounded control is constructed and substantiated for a singularly perturbed optimal elliptic problem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1072–1083, August, 1993.     

Stability of solutions to control constrained nonlinear optimal control problems

We consider a family of nonlinear optimal control problems depending on a parameter. Under the assumption of a second-order sufficient optimality condition it is shown that the solutions of the problems as well as the associated Lagrange multipliers are Lipschitz continuous functions of the parameter.     

On the existence of sporadically catching-up optimal solutions for infinite-horizon optimal control problems

In this paper, we extend the existence theory of Brock and Haurie concerning the existence of sporadically catching-up optimal solutions for autonomous, infinite-horizon optimal control problems. This notion of optimality is one of a hierarchy of types of optimality that have appeared in the literature to deal with optimal control problems whose cost functionals, described by an improper integral, either diverge or are unbounded below. Our results rely on the now classical convexity and seminormality hypotheses due to Cesari and are weaker than those assumed in the work of Brock and Haurie. An example is presented where our results are applicable, but those of the above-mentioned authors do not.This research forms part of the author's doctoral dissertation, written at the University of Delaware, Newark, Delaware, under the supervision of Professor T. S. Angell.     

Existence results without convexity conditions for general problems of optimal control with singular components

This note presents a new, quick approach to existence results without convexity conditions for optimal control problems with singular components in the sense of E. J. McShane (SIAM J. Control5 (1967), 438–485). Starting from the resolvent kernel representation of the solutions of a linear integral equation, a version of Fatou's lemma in several dimensions is shown to lead directly to a compactness result for the attainable set and an existence result for a Mayer problem. These results subsume those of L. W. Neustadt (J. Math. Anal. Appl.7 (1963), 110–117), C. Olech (J. Differential Equations2 (1966), 74–101), M. Q. Jacobs (“Mathematical Theory of Control,” pp. 46–53, Academic Press, 1967), L. Cesari (SIAM J. Control12 (1974), 319–331) and T. S. Angell (J. Optim. Theory Appl.19 (1976), 63–79).