## Computational results of the optimal control of the diffusion equation with the extended conjugate gradient algorithm.(English)Zbl 0961.65064

The authors transform the optimal control problem $J(u,z)= \int^1_0 \int^1_0|z^2(x,t)+ u^2(x,t)|dx dt\to \min$ subject to $$z_t= z_{xx}+ u(x,t)$$ into the unconstrained problem $J(u,z,\mu)= \int^1_0 \int^1_0|z^2(x,t)+ u^2(x,t)|dx dt+\mu\cdot \int^1_0 \int^1_0\|z_t- z_{xx}- u(x,t)\|^2 dx dt\to \min$ and apply the extended conjugate gradient algorithm for the numerical solution. A numerical test is given.

### MSC:

 65K10 Numerical optimization and variational techniques 49M30 Other numerical methods in calculus of variations (MSC2010) 49J20 Existence theories for optimal control problems involving partial differential equations
Full Text:

### References:

 [1] Aderibigbe F.., Ph.D. Thesis (1987) [2] Curtain R.., Functional Analysis in Modern Applied Mathematics (1977) · Zbl 0448.46002 [3] Duchateau ., Partial Differential Equations (1986) [4] Hardorff ., Gradient Optimization and Nonlinear Control (1976) [5] Gelfand, I. and Fomin, S. 1963.Calculus of variations, 226Prentice-Hall. · Zbl 0127.05402 [6] Hestenes M.., Conjugate Direction Methods in Optimization (1980) · Zbl 0439.49001 [7] DOI: 10.1007/BF00939423 · Zbl 0681.90066 [8] DOI: 10.1007/BF00939769 · Zbl 0622.49014 [9] DOI: 10.1016/0022-247X(84)90154-9 · Zbl 0553.49022 [10] Ibiejugba M.., Ph.D. Thesis (1980) [11] Ibiejugba M.., ABACUS 17 pp 19– (1986) [12] Lauwerier H.., Calculus of variation in Mathematical Physics 14 · Zbl 0169.13601 [13] Reju S.., Ph.D. Thesis (1995) [14] DOI: 10.1080/00207169908804836 · Zbl 0937.65069 [15] Reju S.., AMSE J. Modelling 72 (1998) [16] Schuh W.., Optimization and Design of Geodetic Networks pp 185– (1985) [17] Singh M.., SYSTEMS {Decomposition, Optimization and Control) (1978)}
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.