Optimal Control of Partial Differential Equations
Optimal Control of Partial Differential Equations Optimal Control of Partial Differential Equations
118 CHAPTER 10. ALGORITHMS FOR SOLVING OPTIMAL CONTROL PROBLEMS Now, by adding the different equalities, we find the adjoint equation by identifying with ∆t M (C ¯z n − y n d , Cw n u)Y + (D¯z M − yT , Dw M u )YT n=1 ∆t M n=1 More precisely, if p = (p 0 , . . . , p M ) is defined by then (u n , B ∗ p n−1 )U. p M = D ∗ (D¯z M − yT ), for n = 1, . . . , M, p n is the solution to 1 ∆t (−pn + p n−1 ) = A ∗ p n−1 + C ∗ (C ¯z n − y n d ), F ′ M(ū)u = ∆t M n=1 (u n , B ∗ p n−1 )U + ∆t M n=1 (ū n , u n )U. (10.6.17) Observe that the above identification is not justified since D ∗ (D¯z M − yT ) does not necessarily belong to D(A ∗ ). In practice, a ’space-discretization’ is also performed. This means that equation (10.6.15) is replaced by a system of ordinary differential equations, the operator A is replaced by an operator belonging to L(R ℓ ), where ℓ is the dimension of the discrete space, and the above calculations are justified for the corresponding discrete problem. 10.7 Exercises Exercise 10.7.1 Apply the conjugate gradient method to problem (P4) of chapter 5. In particular identify the bounded operator Λ from L 2 (Σ) into L 2 (Ω), and its adjoint Λ ∗ , such that where wu is the solution to equation Exercise 10.7.2 ∂w ∂t |wu(T )| 2 H −1 (Ω) = |Λu| Ω 2 − ∆w = 0 in Q, w = u on Σ, w(0) = 0 in Ω. Apply the conjugate gradient method to problem (P5) of chapter 6. Exercise 10.7.3 Apply the SQP method to problem (P1) of chapter 3. In particular, prove that the Linear- Quadratic problem (QPn+1) of the SQP method is well posed.
Bibliography [1] V. Barbu, Mathematical Methods in Optimization of Differential Systems, Kluwer Academic Publishers, 1994. [2] A. Bensoussan, G. Da Prato, M. C. Delfour, S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Vol. 1, Birkhäuser, 1992. [3] A. Bensoussan, G. Da Prato, M. C. Delfour, S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Vol. 2, Birkhäuser, 1993. [4] H. Brezis, Analyse Fonctionnelle, Theorie et Applications, Masson, Paris, 1983. [5] H. O. Fattorini, Infinite dimensional optimization and control theory, Cambridge University Press, 1999. [6] A. V. Fursikov, Optimal Control of Distributed Systems, Theory and Applications, American Math. Soc., 2000. [7] L. C. Evans, Partial Differential Equations, American Math. Soc., 1999. [8] S. Kesavan, Topics in Functional Analysis and Applications, Wiley-Eastern, New Delhi, 1989. [9] S. Kesavan, Lectures on partial differential equations, FICUS, 2001. [10] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, 1971. [11] J.-L. Lions, Contrôlabilité exacte et Stabilisation des systèmes distribués, Vol. 1, 2, Masson, Paris 1988. [12] J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes, Vol. 1, Dunod, Paris 1968. [13] J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes, Vol. 2, Dunod, Paris 1968. [14] V. Komornik, Exact Controllability and Stabilization, J. Wiley and Masson, Paris,1994. [15] I. Lasiecka, R. Triggiani, Differential and Algebraic Riccati Equations with Applications to Boundary/Point Control Problems: Continuous Theory and Approximation Theory, Springer-Verlag, 1991. 119
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Bibliography<br />
[1] V. Barbu, Mathematical Methods in Optimization <strong>of</strong> <strong>Differential</strong> Systems, Kluwer Academic<br />
Publishers, 1994.<br />
[2] A. Bensoussan, G. Da Prato, M. C. Delfour, S. K. Mitter, Representation and <strong>Control</strong> <strong>of</strong><br />
Infinite Dimensional Systems, Vol. 1, Birkhäuser, 1992.<br />
[3] A. Bensoussan, G. Da Prato, M. C. Delfour, S. K. Mitter, Representation and <strong>Control</strong> <strong>of</strong><br />
Infinite Dimensional Systems, Vol. 2, Birkhäuser, 1993.<br />
[4] H. Brezis, Analyse Fonctionnelle, Theorie et Applications, Masson, Paris, 1983.<br />
[5] H. O. Fattorini, Infinite dimensional optimization and control theory, Cambridge University<br />
Press, 1999.<br />
[6] A. V. Fursikov, <strong>Optimal</strong> <strong>Control</strong> <strong>of</strong> Distributed Systems, Theory and Applications, American<br />
Math. Soc., 2000.<br />
[7] L. C. Evans, <strong>Partial</strong> <strong>Differential</strong> <strong>Equations</strong>, American Math. Soc., 1999.<br />
[8] S. Kesavan, Topics in Functional Analysis and Applications, Wiley-Eastern, New Delhi,<br />
1989.<br />
[9] S. Kesavan, Lectures on partial differential equations, FICUS, 2001.<br />
[10] J.-L. Lions, <strong>Optimal</strong> <strong>Control</strong> <strong>of</strong> Systems Governed by <strong>Partial</strong> <strong>Differential</strong> <strong>Equations</strong>,<br />
Springer, 1971.<br />
[11] J.-L. Lions, Contrôlabilité exacte et Stabilisation des systèmes distribués, Vol. 1, 2, Masson,<br />
Paris 1988.<br />
[12] J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes, Vol. 1, Dunod, Paris<br />
1968.<br />
[13] J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes, Vol. 2, Dunod, Paris<br />
1968.<br />
[14] V. Komornik, Exact <strong>Control</strong>lability and Stabilization, J. Wiley and Masson, Paris,1994.<br />
[15] I. Lasiecka, R. Triggiani, <strong>Differential</strong> and Algebraic Riccati <strong>Equations</strong> with Applications<br />
to Boundary/Point <strong>Control</strong> Problems: Continuous Theory and Approximation Theory,<br />
Springer-Verlag, 1991.<br />
119
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