State analysis and optimal control of linear time-varying systems via Haar wavelets

State analysis and optimization of time-varying systems via Haar wavelets are proposed in this paper. Based upon some useful properties of Haar functions, a special product matrix and a related coefficient matrix are applied to solve the time-varying systems first. Then the backward integration is introduced to solve the adjoint equation of optimization. The unknown Haar coefficient matrix will be in generalized Lyapunov equation form, which is solved via a single-term algorithm. The local property of Haar wavelets is fully applied to shorten the calculation process. A brief comparison between Haar wavelet and other orthogonal functions is also given.