Optimal feedback control for impulsive systems on the space of finitely additive measures.

*(English)*Zbl 1164.34026The paper deals with the existence of measure solutions of the initial value problem for the evolution equation of the form

$$dx=Axdt+f(t,x)dt+g(t,x)\nu \left(dx\right),\phantom{\rule{0.277778em}{0ex}}t\ge 0,\phantom{\rule{0.277778em}{0ex}}x\left(0\right)={x}_{0}\xb7\phantom{\rule{2.em}{0ex}}\left(1\right)$$

Problem (1) is considered in a Banach space $E$. Here, $A$ is the infinitesimal generator of a ${C}_{0}$-semigroup ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ on $E$. Moreover, $f,g:[0,T]\times E\to E$ are measurable functions and $\nu $ is a signed measure. The existence of measure solutions of differential inclusions is also investigated.

Reviewer: Jozef Banaś (Rzeszow)

##### MSC:

34G20 | Nonlinear ODE in abstract spaces |

34A37 | Differential equations with impulses |

34G25 | Evolution inclusions |

49J27 | Optimal control problems in abstract spaces (existence) |

93B52 | Feedback control |