Stabilization of control systems associated with a strongly continuous group
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914929254072320 |
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| author | Nguyen, Hoai-Minh |
| author_facet | Nguyen, Hoai-Minh |
| contents | This paper is devoted to the stabilization of a linear control system $y' = A y + B u$ and its suitable non-linear variants where $(A, \cD(A))$ is an infinitesimal generator of a strongly continuous {\it group} in a Hilbert space $\mH$, and $B$ defined in a Hilbert space $\mU$ is an admissible control operator with respect to the semigroup generated by $A$. Let $λ\in \mR$ and assume that, for some {\it positive} symmetric, invertible $Q = Q(λ) \in \cL(\mH)$, for some {\it non-negative}, symmetric $R = R(λ) \in \cL(\mH)$, and for some {\it non-negative}, symmetric $W = W(λ) \in \cL(\mU)$, it holds $$ A Q + Q A^* - B W B^* + Q R Q + 2 λQ = 0. $$ We then present a new approach to study the stabilization of such a system and its suitable nonlinear variants. Both the stabilization using dynamic feedback controls and the stabilization using static feedback controls in a weak sense are investigated. To our knowledge, the nonlinear case is out of reach previously when $B$ is unbounded for both types of stabilization. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_07560 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stabilization of control systems associated with a strongly continuous group Nguyen, Hoai-Minh Optimization and Control 93B52, 93D15, 93D05, 49J20 This paper is devoted to the stabilization of a linear control system $y' = A y + B u$ and its suitable non-linear variants where $(A, \cD(A))$ is an infinitesimal generator of a strongly continuous {\it group} in a Hilbert space $\mH$, and $B$ defined in a Hilbert space $\mU$ is an admissible control operator with respect to the semigroup generated by $A$. Let $λ\in \mR$ and assume that, for some {\it positive} symmetric, invertible $Q = Q(λ) \in \cL(\mH)$, for some {\it non-negative}, symmetric $R = R(λ) \in \cL(\mH)$, and for some {\it non-negative}, symmetric $W = W(λ) \in \cL(\mU)$, it holds $$ A Q + Q A^* - B W B^* + Q R Q + 2 λQ = 0. $$ We then present a new approach to study the stabilization of such a system and its suitable nonlinear variants. Both the stabilization using dynamic feedback controls and the stabilization using static feedback controls in a weak sense are investigated. To our knowledge, the nonlinear case is out of reach previously when $B$ is unbounded for both types of stabilization. |
| title | Stabilization of control systems associated with a strongly continuous group |
| topic | Optimization and Control 93B52, 93D15, 93D05, 49J20 |
| url | https://arxiv.org/abs/2402.07560 |