Exact and approximate controllability for distributed parameter systems.

*(English)*Zbl 0838.93013
Iserles, A. (ed.), Acta Numerica 1994. Cambridge: Cambridge University Press. 269-378 (1994).

This paper is a very nice exposition of an important set of results in controllability theory. One can learn many useful techniques that occur there, including the Hilbert uniqueness method, the penalty method, optimality conditions, dual problems, approximation methods, dynamic programming. A special emphasis is on numerical approximation to the problems under consideration. Also included are many tables and figures which present the numerical results of test problems. Although the exact controllability is fairly discussed, the focus is the approximate controllability.

Let \(y(t, v)\) be the solution of the state equation corresponding to the control \(v\). Let \(T> 0\) be given and \(y_T\) a given element of the state space. The approximate controllability problem consists in finding \(v\) such that \(y(T, v)\) belongs to \(y_T+ \beta B\), where \(\beta\) is arbitrary small and \(B\) is the unit ball in the state space. To solve this problem, the authors associate two optimal control problems. The first one is \(\inf|v|^2\) over \(v\) in the control space such that \(y(T, v)\in y_T+ \beta B\) while the second one is obtained by penalizing the constraints: \(\inf(|v|^2+ k|y(T, v)- y_T|^2)\) over all \(v\) in the control space. They associate the dual problems which are more appropriate for numerical approximation. The approximation methods combine finite element approximation for the space discretization, finite difference schemes for the time discretization, conjugate gradient algorithms for the solution to discrete problems.

In the first part (for the second part see the review below) the authors focus on distributed and pointwise control systems described by linear diffusion equations.

For the entire collection see [Zbl 0797.00003].

Let \(y(t, v)\) be the solution of the state equation corresponding to the control \(v\). Let \(T> 0\) be given and \(y_T\) a given element of the state space. The approximate controllability problem consists in finding \(v\) such that \(y(T, v)\) belongs to \(y_T+ \beta B\), where \(\beta\) is arbitrary small and \(B\) is the unit ball in the state space. To solve this problem, the authors associate two optimal control problems. The first one is \(\inf|v|^2\) over \(v\) in the control space such that \(y(T, v)\in y_T+ \beta B\) while the second one is obtained by penalizing the constraints: \(\inf(|v|^2+ k|y(T, v)- y_T|^2)\) over all \(v\) in the control space. They associate the dual problems which are more appropriate for numerical approximation. The approximation methods combine finite element approximation for the space discretization, finite difference schemes for the time discretization, conjugate gradient algorithms for the solution to discrete problems.

In the first part (for the second part see the review below) the authors focus on distributed and pointwise control systems described by linear diffusion equations.

For the entire collection see [Zbl 0797.00003].

Reviewer: O.Cârjá (Iaşi)

##### MSC:

93B05 | Controllability |

93C20 | Control/observation systems governed by partial differential equations |

49M05 | Numerical methods based on necessary conditions |