Dynamical systems with an external action (Q1803110)

In a Hilbert space \(H\) a dynamical system of the form (1) \(M\ddot x+u'(x)=f(t,x,x)\), \(x(t_ 0)=x_ 0\), \(x(t_ 0)=y_ 0\) is considered. Here \(M\) is a selfadjoint, bounded and positively defined operator, dot denotes differentiation with respect to the time variable \(t\), comma denotes the Fréchet derivative in \(H\), and \(u\) is the so-called potential functional. Under a number of hypotheses (some of them of technical character) concerning the potential functional \(u\), the function \(f\), and others, the following facts are proven: existence and unicity of the global solution \(x\) of (1) (Lipschitz condition is assumed); boundedness of the total mechanical energy of the solution \(x\) of the system (1): \(H(x,\dot x)=(M\dot x,\dot x)/2+u(x)\) for \(t\) large enough; namely: \(\exists E_{\max}>0\) such that for any trajectory \((x(t),\dot x(t))\) \(\exists t_ 1\geq t_ 0\geq 0\) satisfying \(H(x(t),\dot x(t))<E_{\max}\), if \(t>t_ 1\); (dissipativity of the system).