Dynamics of continuously controlling rod temperature as a singularly perturbed problem
The purpose of this important paper is to study a singularly perturbed problem for the dynamics of continuously controlling rod temperature. The authors consider the following model: \[ \begin{cases} {\partial u\over\partial t}= {\partial^2u\over\partial x^2},\quad u= u(t, x),\quad 0\leq x\leq 1\\ \text{with boundary conditions}\\ u(t,0)= 0,\quad {\partial u(t, 1)\over\partial x}= \sigma\end{cases}\tag{1} \] where the control action \(\sigma= \sigma(t)\) is a solution of the ordinary differential equation: \[ \varepsilon{d\sigma\over dt}= -z+ f(\sigma)\tag{2} \] with a small positive parameter \(\varepsilon\) as a coefficient of the derivative. In the right-hand side of (2), \(z= u(t, x_0)\), where \(0< x_0\leq 1\), and \(f\) is a smooth function \(f(\sigma)= \gamma+ \beta(\sigma- \alpha)[1- (\sigma-\alpha)^2]\), where \(\alpha\), \(\beta\), and \(\gamma\) are positive parameters such that: \(\alpha> 1>\beta\), \(\alpha+ {3-2\beta\over 3\sqrt 3}> \gamma> \alpha\beta\). For \(\varepsilon= 0\) the differential equation (2) becomes an algebraic one and implies that \[ \sigma= \phi_1(z),\quad z\leq z_2,\quad \sigma= \phi_2(z),\quad z\geq z_1\tag{3} \] (\(\phi_1(z)\) and \(\phi_2(z)\) are smooth branches of a multivalued function). Main result: The authors present the existence and stability of a cycle for the relay system (1)--(3) which has an orbitally exponentially stable relaxation cycle for all positive \(\varepsilon\leq\varepsilon_0\).
orbitally exponentially stable relaxation cycle
scientific article; zbMATH DE number 1568237