Sage Journals: Discover world-class research

Abstract

A maximum principle is developed for a class of problems involving the optimal control of a damped parameter system governed by a not-necessarily separable linear hyperbolic equation in two space dimensions. An index of performance is formulated, which consists of functions of the state variable, its first and second order space derivatives and first order time derivative, and a penalty function involving the open-loop control force. The solution of the optimal control problem is shown to be unique using convexity arguments. The maximum principle given involves a Hamiltonian, which contains an adjoint variable as well as an admissible control function. The state and adjoint variables are linked by terminal conditions leading to a boundary/initial/terminal value problem. The maximum principle can be used to compute the optimal control function and is particularly suitable for problems involving the active control of two-dimensional structural elements for vibration suppression.

Keywords

Maximum principle for control damped structures structural control

Get full access to this article

View all access options for this article.

References

Adali, S. , Sadek, I.S. , Sloss, J.M. , and Bruch, J.C., Jr. , 1988, "Distributed control of layered orthotropic plates with damping," Optimal Control Applications and Methods 9(1), 1-17.

Arutyunov, A.U. , Silin, D.B. , and Zerkalov, L.G. , 1992, "Maximum principle and second order conditions for minimax problems of optimal control," Journal of Optimization Theory and Applications 75(3), 521-533.

Basile, N. and Mininni, M. , 1990, "An extension of the maximum principle for a class of optimal control problems in infinite-dimensional space," SIAM Journal on Control and Optimization 28(5), 1113-1135.

Clarke, F. , 1976, "The maximum principle under minimum hypotheses ," SIAM Journal on Control and Optimization 14(6), 1078-1091.

Fattorini, H.D. , 1985, "The maximum principle for nonlinear, nonconvex systems in infinite dimensional spaces," Lecture Notes in Control and Information Sciences 75, Springer-Verlag , New York, 162-178.

Gregson, M.J. , 1973, "A maximum principle for the optimal control of distributed parameter systems," in Recent Matheinatical Developments in Control, D. J., Bell , ed., Academic Press, New York, 309-321.

Kaskosz, B. , 1990, "A maximum principle in relaxed controls," Nonlinear Analysis—Theory, Methods and Applications 14(4), 357-367.

Li, X. and Yong, J. , 1981, "Necessary conditions for optimal control of distributed parameter systems," SIAM Journal on Control and Optimization 29(4), 895-908.

Sadigh-Esfandiari, R. , Sloss, J.M. , and Bruch, J.C., Jr. , 1990, "Maximum principle for optimal control of distributed parameter systems with singular mass matrix," Journal of Optimization Theory and Applications 66(2), 211-226.

10.

Sloss, J.M. , Bruch, J.C., Jr. , and Sadek, I.S. , 1989, "A maximum principle for nonconservative self-adjoint systems," IMA Journal of Mathematical Control and Information 6, 199-216.

11.

Sloss, J.M. , Sadek, I.S. , Bruch, J.C., Jr. , and Adali, S. , 1989a, "Applications of a maximum principle for the structural control of laminated composite plates," IMA Journal of Mathematical Control and Information 6, 217-232.

12.

Sloss, J.M. , Sadek, I.S. , Bruch, J.C., Jr. , and Adali, S. , 1989b, "Optimal vibration control of laminated cross-ply cylindrical panels," Mathematical Engineering in Industry 2(3), 169-188.

13.

Sloss, J.M. , Sadek, I.S. , Bruch, J.C., Jr. , and Adali, S. , 1995, "Maximum principle for the optimal control of a hyperbolic equation in one space dimension, part 1: theory," Journal of Optimization Theory and Applications 87(1), 33-45.

Maximum Principle for the Optimal Control of a Hyperbolic Equation in Two Space Dimensions

Abstract

Keywords

Get full access to this article

References