Stability and maximum boundedness of processes in systems with pulse- width modulation (Q1115398)

Consider the integral equation \[ \sigma(t)= \alpha_ 0+\alpha(t)+\psi(t)-\int^{t}_{0} [\rho+\gamma(t-\tau)] \phi(\tau)d\tau \] where \(\phi(t)=sgn \sigma(nT)\) \(nT\leq t\leq nT+\tau_ n\), \(\phi(t)=0\), \(nT+\tau_ n<t\leq (n+1)T\) and \(\tau_ n\) is defined by some nonlinear equation. Here \(\gamma (\cdot)\in L^ 1(0,\infty),\) \(| {\dot \gamma}(t)| +{\ddot \gamma}(t)| \leq c_ 0e^{- \alpha t}\) and \(\psi(\cdot)\) is a piecewise continuous function. Frequency domain conditions expressed in terms of \({\tilde \gamma}(\sigma)\), the Laplace transform of \(\gamma(t)\), are obtained for the asymptotic behaviour of the difference of two arbitrary solutions of the integral equation as well as for dissipativity (ultimate boundedness) of the solutions.