Benchmarks

As the state $u\_s$ depends on control $f\_s$ , then $u\_s$ can also be written as $u\_s=u\_s(f\_s)$ and so from now we can use notation $J\_s(f\_s)$ instead of $J\_s(u\_s(f\_s),f\_s)$ . In the above problem 3.1,3.2,3.3 we are going to find an optimal control $\backslash bar\{f\}\_s\backslash in\; U\_\{ad\}\backslash subset\; L^2(\backslash Omega)$ in such a way that the corresponding solution of $\backslash bar\{u\}\_s$ together with $\backslash bar\{f\}\_s$ satisfies the minimization of the cost function, i.e.,

$$J\_s(\backslash bar\{f\}\_s):=\backslash min\_\{f\_s\backslash in\; U\_\{ad\}\}J\_s(f\_s),$$

and corresponding optimal control of classical Poisson equation with homogeneous boundary conditions is the following

$$\backslash min\; J(u,f)=\backslash frac\{1\}\{2\}\backslash left(\backslash |\backslash nabla\; u\backslash |\_\{L^2(\backslash Omega)\}^2+\backslash mu\backslash |f\backslash |\_\{L^2(\backslash Omega)\}^2\backslash right)\; \backslash tag\{3.4\}$$

subject to (the Poisson equation )

and the control constraints

$$a\backslash le\backslash |f\backslash |\_\{L^2(\backslash Omega)\}\backslash le\; b.\; \backslash tag\{3.6\}$$

Our main objective is to find the optimal control $\backslash bar\{f\}$ such that

$$J(\backslash bar\{f\})=\backslash min\_\{f\backslash in\; U\_\{ad\}\}J(f).$$

Proposition 3.1 (see [3]). Let be the sequence satisfying $\backslash |f\_s\backslash |\_\{H^\{-s\}(\backslash Omega)\}$ uniformly bounded with respect to $s$ and $f\_s\backslash to\; f$ weakly in $H^\{-1\}(\backslash Omega)$ as $s\backslash to\; 1^-$ , then $u\_s\backslash to\; u$ strongly in $H\_0^\{1-\backslash delta\}(\backslash Omega)$ for some $C>0$ and .

Here, we will extend this proposition 2.2 to optimal control of fractional PDE in our main result.

The first proposition represents about the Poincaré inequality in fractional Sobolev space and the second refers to the minimizer of a function defined on a suitable space.

Proposition 3.2 (Poincaré inequality, [4]). Let $s\backslash in(0,1)$ , $\backslash Omega\backslash subset\backslash mathbb\{R\}^N$ be an open and bounded set then we have

$$\backslash |u\backslash |\_2^2\backslash le\; C(N,\backslash Omega,s)[u]\_\{H^s(\backslash Omega)\}^2.$$

where some constant $C(N,\backslash Omega,s)$ depending upon $N$ , $\backslash Omega$ and $s$ .

Proposition 3.3. Let the admissible control space $U\_\{ad\}$ be weakly closed, bounded subset of $L^2(\backslash Omega)$ with $J\_s:U\_\{ad\}\backslash to\backslash mathbb\{R\}$ is weakly lower semi-continuous. Then $J\_s$ has minimizer in $U\_\{ad\}$ .

The existence and uniqueness of optimal control via strict convex and lower semi-continuous is discussed in the following proposition.

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