Sharp error estimates for a finite element-penalty approach to a class of regulator problems

Authors:
Goong Chen, Wendell H. Mills, Shun Hua Sun and David A. Yost

Journal:
Math. Comp. **40** (1983), 151-173

MSC:
Primary 65K10; Secondary 49D30

DOI:
https://doi.org/10.1090/S0025-5718-1983-0679438-1

MathSciNet review:
679438

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Abstract: Quadratic cost optimal controls can be solved by penalizing the governing linear differential equation [2], [9]. In this paper, we study the numerical analysis of this approach using finite elements. We formulate the geometric *condition* (H) which requires that pairs of certain related finite-dimensional approximation spaces form "angles" which are bounded away from the "180 angle". Under condition (H), we prove that the penalty parameter and the discretization parameter *h* are independent in the error bounds, thereby giving sharp asymptotic error estimates. This condition (H) is shown to be also a necessary condition for such independence. Examples and numerical evidence are also provided.

**[1]**A. K. Aziz (ed.),*The mathematical foundations of the finite element method with applications to partial differential equations*, Academic Press, New York-London, 1972. MR**0347104****[2]**A. V. Balakrishnan,*On a new computing technique in optimal control*, SIAM J. Control**6**(1968), 149–173. MR**0250154****[3]**W. E. Bosarge Jr. and O. G. Johnson,*Error bounds of high order accuracy for the state regulator problem via piecewise polynomial approximations*, SIAM J. Control**9**(1971), 15–28. MR**0289179****[4]**Goong Chen and Wendell H. Mills Jr.,*Finite elements and terminal penalization for quadratic cost optimal control problems governed by ordinary differential equations*, SIAM J. Control Optim.**19**(1981), no. 6, 744–764. MR**634952**, https://doi.org/10.1137/0319049**[5]**F. Deutsch, "The alternating method of von-Neumann," in*Multivariate Approximation Theory*(W. Schempp and K. Zeller, eds.), Birkhäuser Verlag, Basel, 1979.**[6]**Richard S. Falk,*A finite element method for the stationary Stokes equations using trial functions which do not have to satisfy 𝑑𝑖𝑣𝜈=0*, Math. Comp.**30**(1976), no. 136, 698–702. MR**421109**, https://doi.org/10.1090/S0025-5718-1976-0421109-7**[7]**Richard S. Falk and J. Thomas King,*A penalty and extrapolation method for the stationary Stokes equations*, SIAM J. Numer. Anal.**13**(1976), no. 5, 814–829. MR**471382**, https://doi.org/10.1137/0713064**[8]**I. C. Gohberg and M. G. Kreĭn,*Introduction to the theory of linear nonselfadjoint operators*, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. MR**0246142****[9]**J.-L. Lions,*Optimal control of systems governed by partial differential equations.*, Translated from the French by S. K. Mitter. Die Grundlehren der mathematischen Wissenschaften, Band 170, Springer-Verlag, New York-Berlin, 1971. MR**0271512****[10]**B. T. Polyak, "The convergence rate of the penalty function method,"*Zh. Vychisl. Mat. i Mat. Fiz.*, v. 11, 1971, pp. 3-11. (Russian)**[11]**David L. Russell,*Mathematics of finite-dimensional control systems*, Lecture Notes in Pure and Applied Mathematics, vol. 43, Marcel Dekker, Inc., New York, 1979. Theory and design. MR**531035****[12]**Gilbert Strang and George J. Fix,*An analysis of the finite element method*, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1973. Prentice-Hall Series in Automatic Computation. MR**0443377**

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DOI:
https://doi.org/10.1090/S0025-5718-1983-0679438-1

Article copyright:
© Copyright 1983
American Mathematical Society