Dissipative distributed systems (Q2722570)

This paper deals with distributed dynamical systems described by constant-coefficient Partial Differential Equations (PDEs).NEWLINENEWLINENEWLINEA distributed differential system is a triple \((\mathbb{R}^n, \mathbb{R}^w,{\mathcal B})\), with behavior \({\mathcal B}\) consisting of the solution set of a system of PDEs NEWLINE\[NEWLINER\Biggl({\partial\over\partial x_1},\dots, {\partial\over\partial x_n}\Biggr) w= 0\tag{1}NEWLINE\]NEWLINE viewed as an equation in the functions \((x_1,\dots, x_n)= x\in\mathbb{R}^n\mapsto (w_1(x),\dots, w_w(x))= w(x)\in \mathbb{R}^w\), where \(R\in\mathbb{R}^{0\times w}[\xi_1,\dots, \xi_n]\) is a matrix of polynomials in \(\mathbb{R}[\xi_1,\dots, \xi_n]\), and the behavior of this system of PDEs is defined as NEWLINE\[NEWLINE{\mathcal B}=\{w\in C^\infty(\mathbb{R}^n, \mathbb{R}^w)\mid(1)\text{ is satisfied}\}.NEWLINE\]NEWLINE More specifically, one studies conservative systems, where for compact support trajectories the integral of the supply rate is zero, and the dissipative systems, respectively, for which the integral is nonnegative. The main interest is to express a global conservation or dissipation law as a local one, involving the flux density and the dissipation rate. Concerning the construction of the flux density and the dissipation rate there exists a relation with Hilbert's 17th problem on the factorization of real nonnegative rational functions in many variables as a sum of squares of real functions. On the other hand, the local conservation or dissipation laws necessarily involve `hidden' latent variables.NEWLINENEWLINEFor the entire collection see [Zbl 0961.00036].