The linear quadratic optimal control problem for hereditary differential systems: Theory and numerical solution

In the first part of this paper, we summarize and complete earlier results of Delfour-Mitter [3] on the optimal control problem for linear hereditary differential systems (HDS) with a linear-quadratic cost function. The properties of the operator Π(t) which characterizes the feedback gains and the reference functionr(t) are fully detailed. In a second part, we construct an approximation to the linear HDS in state form and to the linear adjoint state equation and prove convergence. In a third part we construct and solve the approximate optimal control problem following the method of J. C. Nédelec. In the last part we construct the approximation to Π(t) andr(t) and prove convergence. Finally we give a number of typical examples to illustrate the main features of the kernel of Π(t).

[1]  H. Banks,et al.  Optimal control of linear time-delay systems , 1969 .

[2]  Harvey Thomas Banks,et al.  Projection series for retarded functional differential equations with applications to optimal control problems , 1975 .

[3]  P. Sannuti,et al.  Asymptotic series solution of optimal systems with small time delay , 1973 .

[4]  E. Lee,et al.  The quadratic problem for systems with time delays , 1971 .

[5]  On Some Functional Differential Equations: Existence of Solutions and Difference Approximations , 1968 .

[6]  Alan Feldstein,et al.  Numerical Methods for Nonlinear Volterra Integro-Differential Equations , 1974 .

[7]  N. N. Krasovskii,et al.  The approximation of a problem of analytic design of controls in a system with time-lag , 1964 .

[8]  Robert E. Kalaba,et al.  Numerical integration of a differential-difference equation with a decreasing time-lag , 1965, CACM.

[9]  William R. Perkins,et al.  Optimization of time delay systems using parameter imbedding , 1973 .

[10]  Moustafa A. Soliman,et al.  Optimal feedback control for linear-quadratic systems having time delays† , 1972 .

[11]  Michel C. Delfour,et al.  Controllability, Observability and Optimal Feedback Control of Affine Hereditary Differential Systems , 1972 .

[12]  M. Jacobs,et al.  An optimum settling problem for time lag systesm , 1972 .

[13]  Hitoshi Sasai,et al.  On the Convergence of Approximating Solutions for Linear Distributed Parameter Optimal Control Problems , 1971 .

[14]  J. Lions,et al.  Problèmes aux limites non homogènes et applications , 1968 .

[15]  H. Trotter.  Approximation of semi-groups of operators , 1958 .

[16]  D. W. Ross,et al.  An Optimal Control Problem for Systems with Differential-Difference Equation Dynamics , 1969 .

[17]  Convergence of difference methods for initial and boundary value problems with discontinuous data , 1971 .

[18]  D. Ross.  Controller design for time lag systems via a quadratic criterion , 1971 .

[19]  Moustafa A. Soliman,et al.  Optimal control of multivariable systems with pure time delays , 1971 .

[20]  Richard Goodman,et al.  Numerical solution of ordinary and retarded differntial equations with discontinuou derivatives , 1973 .

[21]  Harold J. Kushner,et al.  On the Control of a Linear Functional Differential Equation with Quadratic Cost , 1970 .

[22]  Peddapullaiah Sannuti,et al.  Optimal control of singularly perturbed time delay systems with an application to a coupled core nuclear reactor , 1974, CDC 1974.

[23]  Iu M Repin,et al.  On the approximate replacement of systems with lag by ordinary dynamical systems , 1965 .

[24]  M. Delfour.  Linear Hereditary Differential Systems and their Control , 1974 .

[25]  Peter Linz,et al.  Linear Multistep Methods for Volterra Integro-Differential Equations , 1969, JACM.

[26]  J. A. Burns,et al.  An Abstract Framework for Approximate Solutions to Optimal Control Problems Governed by Hereditary Systems , 1975 .

[27]  M. N. Spijker.  Convergence and stability of step-by-step methods for the numerical solution of initial-value problems , 1966 .

[28]  J. Lions.  Optimal Control of Systems Governed by Partial Differential Equations , 1971 .

[29]  Andrzej P. Wierzbicki,et al.  Computational Methods in Hilbert Space for Optimal Control Problems with Delays , 1973, Optimization Techniques.

[30]  C. Cryer,et al.  The Numerical Solution of Volterra Functional Di erential Equations by Euler''s Method , 1972 .

[31]  Michel C. Delfour,et al.  Hereditary Differential Systems with Constant Delays. I. General Case , 1972 .

[32]  Jack K. Hale,et al.  LINEAR FUNCTIONAL-DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS , 1963 .

[33]  Hitoshi Sasai,et al.  Consideration of Linear Delay-Differerential Systems by Approximation Systems , 1974 .