Nonlinear fuzzy differential equation with time delay and optimal control problem (Q304930)

The author proves the existence of a mild fuzzy solution to fuzzy systems NEWLINE\[NEWLINEDx (t)=a(t)x(t)+f(t,x(t)),\quad x(0)=x_0,\quad t\in [0,T], \eqno{(1)}NEWLINE\]NEWLINE NEWLINE\[NEWLINEDx(t)=a(t)x(t)+g(t,x(t),x_t),\quad t\in [0,T],\quad x(t)=\phi (t),\quad t\in [-l,0], \eqno{(2)}NEWLINE\]NEWLINE NEWLINE\[NEWLINEDx(t)=a(t)x(t)+g(t,x(t),x_t,u(t)),\quad t\in [0,T],\quad x(t)=\phi (t),\quad t\in [-l,0], \eqno{(3)}NEWLINE\]NEWLINE where \(x(t)\) is a fuzzy state function of time variable \(t\), \(f(t,x)\) is a fuzzy input function of the variables \(t\) and \(x\), \(Dx\) is the fuzzy Hukuhara derivative of \(x\), \(x(0)=x_0\) is a fuzzy number, \(a:[0,T]\to \mathbb R\) is a continuous function, \(x_t=x(t+\theta)\) for \(-l\leq\theta\leq 0\) is the state that is time-delayed, \(g(t,x,x_t)\) is a fuzzy input function of the variables \(t\), \(x\) and \(x_t\), \(\phi\) is a fuzzy history function before the start time \(t=0\), \(u\) is a fuzzy controller function of time variable \(t\).NEWLINENEWLINEAlso, the author proves the existence of a solution to the problem NEWLINE\[NEWLINEJ(x,u)=\int_0^Tr(t,x,x_t,u(t))dt+k(x(T))NEWLINE\]NEWLINE constrained by system (3), where \(r\) and \(k\) are real functions, i.e. \(J(x,u)\in \mathbb R\).