Impulse dynamic systems and applications to mathematical economics. (English) Zbl 0805.34009

The differential equation of the form (1) \(dx/dt= f(x) + B(x,v) \cdot dv/dt\), \(x(0) = x_ 0\), is considered. Here \(f : \mathbb{R}^ n \to \mathbb{R}^ n\), \(B\) is an \((m \times n)\)-matrix function and \(v : \mathbb{R} \to \mathbb{R}^ m\) is a time program. Suggesting the procedure for the multiplication of the discontinuous function \(B(x(\cdot)\), \(v(\cdot))\) and the impulse function \(Dv\) the author gives the definition and the description for a weak solution of the equation (1). This approach allows to consider some mathematical market models with discontinuous current prices and to discuss stability and optimization problems.


34A37 Ordinary differential equations with impulses
91B50 General equilibrium theory