## Nonlinear fuzzy differential equation with time delay and optimal control problem.(English)Zbl 1348.34005

The author proves the existence of a mild fuzzy solution to fuzzy systems $Dx (t)=a(t)x(t)+f(t,x(t)),\quad x(0)=x_0,\quad t\in [0,T], \eqno{(1)}$
$Dx(t)=a(t)x(t)+g(t,x(t),x_t),\quad t\in [0,T],\quad x(t)=\phi (t),\quad t\in [-l,0], \eqno{(2)}$
$Dx(t)=a(t)x(t)+g(t,x(t),x_t,u(t)),\quad t\in [0,T],\quad x(t)=\phi (t),\quad t\in [-l,0], \eqno{(3)}$ 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 $$t$$.
Also, the author proves the existence of a solution to the problem $J(x,u)=\int_0^Tr(t,x,x_t,u(t))dt+k(x(T))$ constrained by system (3), where $$r$$ and $$k$$ are real functions, i.e. $$J(x,u)\in \mathbb R$$.

### MSC:

 34A07 Fuzzy ordinary differential equations 93C42 Fuzzy control/observation systems 34K36 Fuzzy functional-differential equations 49J15 Existence theories for optimal control problems involving ordinary differential equations

### Keywords:

fuzzy system; mild solution; control; Hukuhara; time delay
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### References:

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