Gap Phenomenon in Modeling of Suboptimal Controls to Parabolic Optimal Control Problems in Thick Multistructures

Abstract

In this chapter, we study the asymptotic behavior of the following class of the parabolic optimal control problems (OCPs)

$$I_{\varepsilon}(u_{\varepsilon},y_{\varepsilon})=\int_0^T\int_{\Omega^{+}}(y_{\varepsilon}- q_0)^2 \,\mathrm{d} x\,\mathrm{d} t \, +\,\int_0^T\int_{\Gamma_{\varepsilon}}u^2_{\varepsilon}\,\mathrm{d} x^\prime \,\mathrm{d} t \longrightarrow \inf,$$

$$\left.\begin{array}{rcll}y^\prime_{\varepsilon}-\Delta_x \ y_{\varepsilon}+y_{\varepsilon}& = & f_{\varepsilon} &\quad \text{ in} \ (0,T)\times \Omega_{\varepsilon},\\\partial_{\nu} y_{\varepsilon}& = & - \varepsilon \,k_0y_{\varepsilon}& \quad \text{ on } \ (0,T)\times S_{\varepsilon},\\y_{\varepsilon}& = & u_{\varepsilon}& \quad \text{ on } \ (0,T)\times \Gamma_{\varepsilon},\\\partial_{\nu} y_{\varepsilon}& = & 0 & \quad \text{ on } \ (0,T)\times \partial\Omega_{\varepsilon}\setminus \left(\Gamma_{\varepsilon}\cup S_{\varepsilon}\right),\\y_{\varepsilon}(0,x) & = & y^0_{\varepsilon}& \quad \text{ a.e. } \ x\in \Omega_{\varepsilon}\end{array}\right\}$$

as a small parameter ε tends to 0.