An optimal control problem with unbounded control operator and unbounded observation operator where the Algebraic Riccati Equation is satisfied as a Lyapunov equation
✍ Scribed by R. Triggiani
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 299 KB
- Volume
- 10
- Category
- Article
- ISSN
- 0893-9659
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✦ Synopsis
We provide an optimal control problem for a one-dimensional hyperbolic equation over = (0, c~), with Dirichlet boundary control u(t) at x = 0, and point observation at x = 1, over an infinite time horizon. Thus, both control and observation operators B and R are unbounded. Because of the finite speed of propagation of the problem, the initial condition yo(x) and the control u(t) do not interfere. Thus, the optimal control u°(t) = 0. A double striking feature of this problem is that, despite the unboundedness of both B and R, (i) the (unbounded) gain operator B*P vanishes over T)(A), A being the basic (unbounded) free dynamics operator, and (ii) the Algebraic Riccati Equation is satisfied by P on T)(A), indeed as a Lyapunov equation (linear in P).