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

Part of the Systems & Control: Foundations & Applications book series (SCFA)

## 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.

## Keywords

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