The steepest descent dynamical system with control. Applications to constrained minimization

Cabot, Alexandre

ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 2, pp. 243-258

Let $H$ be a real Hilbert space, ${\Phi }_1:H\to ℝ$ a convex function of class ${𝒞}^1$ that we wish to minimize under the convex constraint $S$. A classical approach consists in following the trajectories of the generalized steepest descent system (cf. Brézis [5]) applied to the non-smooth function ${\Phi }_1+{\delta }_S$. Following Antipin [1], it is also possible to use a continuous gradient-projection system. We propose here an alternative method as follows: given a smooth convex function ${\Phi }_0:H\to ℝ$ whose critical points coincide with $S$ and a control parameter $\epsilon :{ℝ}_{+}\to {ℝ}_{+}$ tending to zero, we consider the “Steepest Descent and Control” system

$$\left(SDC\right)\dot{x}\left(t\right)+\nabla {\Phi }_0\left(x\left(t\right)\right)+\epsilon \left(t\right)\nabla {\Phi }_1\left(x\left(t\right)\right)=0,$$

where the control $\epsilon$ satisfies ${\int }_0^{+\infty }\epsilon \left(t\right)\mathrm{d}t=+\infty$. This last condition ensures that $\epsilon$ “slowly” tends to $0$. When $H$ is finite dimensional, we then prove that $d(x\left(t\right),{argmin}_S{\Phi }_1)\to 0(t\to +\infty ),$ and we give sufficient conditions under which $x\left(t\right)\to \overline{x}\in {argmin}_S{\Phi }_1$. We end the paper by numerical experiments allowing to compare the $\left(SDC\right)$ system with the other systems already mentioned.

Classification : 34A12, 34D05, 34G20, 34H05, 37N40
Keywords: dissipative dynamical system, steepest descent method, constrained optimization, convex minimization, asymptotic behaviour, non-linear oscillator

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