An optimal control problem for a parabolic equation in non-cylindrical domains
The authors study an optimal control problem for a parabolic equation with homogeneous Dirichlet data described by \[ (1)\quad (\partial /\partial t)y(t,x)=\Delta y(t,x)+u(t,x),\quad t\geq 0,\quad x\in \Omega_ t, \] \[ y(0,x)=y_ 0(x),\quad x\in \Omega_ 0,\quad y(t,x)=0,\quad t\geq 0,\quad x\in \Gamma_ t. \] Here, \(\Omega_ t\) is a bounded open set of \({\mathbb{R}}^ n\) with smooth boundary \(\Gamma_ t\); y is the state, and u the control belonging to \(L^ 2([0,\infty)\times \Omega_ t)\). The problem is to find a u minimizing the cost \[ J(u)=\int^{\infty}_{0}dt\int_{\Omega_ t}(| y|^ 2+| u|^ 2)dx. \] The point here is that equation (1) is written in the non-cylindrical domain \(U_{t>0}(t\times \Omega_ t)\). Equation (1) is, however, transformed into a standard equation in a cylindrical domain by change of variables: A system of ordinary differential equations is considered: \[ (2)\quad d\beta /dt=V(t,\beta),\quad \beta (0)=x\in {\mathbb{R}}^ n. \] An assumption on the function V is that the solution to (2), denoted by \(T_ t(x)\), satisfies \(T_ t(\Omega_ 0)=\Omega_ t\) and \(T_ t(\Gamma_ 0)=\Gamma_ t\), \(t\geq 0\). By setting \(z(t,X)=y(t,T_ t(X))\) and \(v(t,X)=u(t,T_ t(X))\), equation (1) is transformed into an equation in \([0,\infty)\times \Omega_ 0\), or an abstract equation in \(L^ 2(\Omega_ 0)\); \[ (3)\quad dz/dt=B(t)z+v,\quad z(0)=y_ 0. \] Here, B(t) is derived from \(\Delta\), and \(D(B(t))=H^ 2(\Omega_ 0)\cap H^ 1_ 0(\Omega_ 0)\). Then, the original problem is equivalent to finding a v minimizing \[ \bar J(v)=\int^{\infty}_{0}dt\int_{\Omega_ 0}| \det DT_ t| (| z|^ 2+| v|^ 2)dx, \] \(DT_ t\) being the Jacobian matrix of \(T_ t\). The other assumption is the existence of a bounded domain \(\sigma\) such that \(\Omega_ t\subset \sigma\), \(t\geq 0\) in order to ensure admissible controls. At this stage, it is standard to solve the optimal control problem for equation (3). As usual, the Riccati equation associated with (3) is solved. The unique optimal control \(v'=v^*\) is found in the feedback form. The solution to the original optimal control problem is finally stated via the mapping \(T_ t\).
parabolic equation with homogeneous Dirichlet data
scientific article; zbMATH DE number 4071755
scientific article