An optimal control problem for a parabolic equation in non-cylindrical domains
type: Item
instance of: scholarly article
title: An optimal control problem for a parabolic equation in non-cylindrical domains
zbMATH Open document ID: 0656.49001
DOI: 10.1016/0167-6911(88)90114-4
published in: Systems \& Control Letters
publication date: 1988-01-01T00:00:00Z
review text: 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 con…
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 35B37
Mathematics Subject Classification ID: 93C05
zbMATH DE Number: 4071755
zbMATH Keywords: optimal control
zbMATH Keywords: parabolic equation with homogeneous Dirichlet data
zbMATH Keywords: non- cylindrical domain
zbMATH Keywords: Riccati equation
zbMATH Keywords: feedback
author: Guiseppe Da Prato
author: J. P. Zolesio
MaRDI profile type: Publication
full work available at URL: 0167-6911(88)90114-4
OpenAlex ID: W2155192930
cites work: Quadratic Control for Linear Time-Varying Systems
cites work: Q5549989
cites work: Q3219250
cites work: Q3666416
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description: scientific article; zbMATH DE number 4071755