Minimal time impulse control for a class of homogeneous evolution equations (Q6604019)

Let \(X\) and \(U\) be two real separable Hilbert spaces which are the state space and the control space. The authors consider the impulse controlled homogeneous evolution equations at a given time \(\tau \geq 0\): \(x^{\prime }(t)=Ax(t)+g(t)x(t)\), \(t\in (0,+\infty )\setminus \{\tau \}\), \(x(0)=x_{0}\), \( x(\tau )=x(\tau ^{-})+Bu\), where \(x_{0}\in X\) is a given initial state, \( u\in U\) is a control input, \(B\in \mathcal{L}(U,X)\), \(x(\cdot )\) is right continuous at \(\tau \), \(x(\tau ^{-})=\lim_{t\rightarrow \tau ^{-}}x(t)\) in \(X\), \(A:D(A)\subseteq X\rightarrow X\) is the infinitesimal generator of a \(C^{0} \)-semigroup such that \(e^{tA}\) is compact for \(t>0\), \(B\in \mathcal{L}(U,X)\) and \(B\) is compact. \N\NA function \(x:[0,\infty )\rightarrow X\) is a mild solution to the above problem if it satisfies: \(x(t)=e^{tAx_{0}}+ \int_{0}^{t}e^{(t-s)Ag(s)}x(s)ds\), if \(t\in \lbrack 0,\tau )\), and \( x(t)=e^{tAx_{0}}+e^{(t-\tau )A}Bu+\int_{0}^{t}e^{(t-s)Ag(s)}x(s)ds\), if \(t\in \lbrack \tau ,+\infty )\). A mild solution \(x^{\tau }(\cdot ;x_{0},u)\) to the above problem satisfies \(x^{\tau }(\cdot ;x_{0},u)\in C([0,\tau );X)\cup C([\tau ,T];X)\) for every \(T>\tau \geq 0\). The authors define the set \(\mathfrak{U}_{ad}\) of admissible controls as: \(\mathfrak{U}_{ad}=\{u\in \mathfrak{U}_{M}:x^{\tau }(T;x_{0},u)\in B_{r}(0)\) for some \(T\geq \tau \}\), where \(\mathfrak{U}_{M}=\{u\in U:\left\Vert u\right\Vert _{U}\leq M\}\) and \( B_{r}(0)=\{x\in X:\left\Vert x\right\Vert _{X}\leq r\}\) is the target set. The optimal time impulse control problem is defined as: \(t^{\ast }(M,\tau )=inf\{T\geq \tau :x^{\tau }(T;x_{0},u)\in B_{r}(0)\) and \(u\in \mathfrak{U} _{M}\}\). An admissible control \( u^{\ast }\) is called the optimal control if \(x^{\tau }(t^{\ast }(M,\tau );x_{0},u^{\ast })\in B_{r}(0)\). \N\NThe first main result proves under hypotheses that the optimal time impulse control problem has a unique optimal control \(u^{\ast }\). Moreover, \(t^{\ast }(M,\tau )>\tau \) and \(\left\Vert u^{\ast }\right\Vert _{U}=M\). The second main result proves that then the function \(t^{\ast }(\cdot ,\cdot )\) is continuous at the point \((M,\tau )\). The paper ends with the analysis of two examples.