Abstract
We consider first nonlinear systems of the form
x=A(x)x+B(x)u
together with a standard quadratic cost functional and replace the system by a sequence of time-varying approximations for which the optimal control problem can be solved explicitly. We then show that the sequence converges. Although it may not converge to a global optimal control of the nonlinear system, we also consider a similar approximation sequence for the equation given by the necessary conditions of the maximum principle and we shall see that the first method gives solutions very close to the optimal solution in many cases. We shall also extend the results to parabolic PDEs which can be written in the above form on some Hilbert space.
Similar content being viewed by others
References
Z. Aganovic and Z. Gajic, The successive approximation procedure for finite time optimal control of bilinear systems, IEEE Trans. Automat. Control 39(9) (1994) 324–328.
H. Amann, Periodic solutions of semilinear parabolic equations, in: Nonlinear Analysis, A collection of papers in honor of Erich Roth (Academic Press, New York, 1978) pp. 1–29.
S.P. Banks and M.K. Yew, On a class of suboptimal controls for infinite-dimensional bilinear systems, Systems Control Lett. 5 (1985) 327–333.
S.P. Banks, On the optimal control of nonlinear systems, Systems Control Lett. 6 (1986) 337–343.
S.P. Banks, Infinite-dimensional Carleman linearisation, the Lie series and optimal control of nonlinear PDEs, Internat. J. Systems Sci. 23 (1992) 663–675.
S.P. Banks and K. Mhana, Optimal control and stabilization for nonlinear systems, IMA J. Control and Inform. 9 (1992) 179–196.
S.P. Banks and S.K. Al-Jurani, Lie algebras and the stability of nonlinear systems, Internat. J. Control 60 (1994) 315–329.
S.P. Banks, A. Moser and D. McCaffrey, Clifford algebras, dynamical systems and periodic orbits, IMA J. Math. Control and Inform. 13 (1996) 279–298.
S.P. Banks and D. McCaffrey, Optimal control of nonlinear distributed parameter systems, Proc. of 17th IMACS World Congress, Berlin (July 1997).
S.P. Banks and D. McCaffrey, Lie algebras, structure of nonlinear systems and chaotic motion, Internat. J. Bifur. and Chaos 8(7) (1998) 1437–1462.
S.P. Banks, A. Moser and D. McCaffrey, Robust exponential stability of evolution equations, Arch. Control Sci. 4(XL) (1995) 261–279.
F. Brauer, Perturbations of nonlinear systems of differential equations, J. Math. Anal. App. 14 (1966) 198–206.
R.F. Curtain and A.J. Pritchard, Infinite-Dimensional Linear Systems Theory (Springer, 1978).
A. Friedman, Partial Differential Equations (Holt, New York, 1969).
W.L. Garrard and J.M. Jordan, Design of nonlinear automatic flight control systems, Automatica 13 (1977) 497–505.
J.K. Hale, Theory of Functional Differential Equations (Springer, New York, 1977).
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840 (Springer, 1981).
S.K. Nguang, Robust nonlinear H 1-output feedback control, IEEE Trans. Automat. Control 41(7) (1996) 1003–1007.
S.M. Roberts and J.S. Shipman, Two-Point Boundary Value Problems: Shooting Methods, Modern Analytic and Computational Methods in Science and Mathematics 31, ed. R. Bellman (Elsevier, New York, 1972).
P. Soravia, H 1 control of nonlinear systems: Differential games and viscosity solutions, SIAM J. Control Optim. 34(3) (1996) 1071–1097.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Banks, S., Dinesh, K. Approximate Optimal Control and Stability of Nonlinear Finite- and Infinite-Dimensional Systems. Annals of Operations Research 98, 19–44 (2000). https://doi.org/10.1023/A:1019279617898
Issue Date:
DOI: https://doi.org/10.1023/A:1019279617898