On construction of Lyapunov functions for scalar linear time-varying systems

It is known that the construction of Lyapunov functions for scalar linear time-varying systems is related with solutions to the scalar Lyapunov differential equation, whose solution involves both improper integrals and double integrals, and thus are not easy to compute in general. This paper establishes a systematic method for constructing Lyapunov functions for scalar linear time-varying systems. The constructed Lyapunov functions involve an integral of the system parameter with a weighting function over a finite interval. Explicit conditions are imposed on the weighting function and the integral interval such that the Lyapunov function is both positive definite and uniformly bounded, and its time-derivative is negative definite. As a result, constructive solutions to the associated scalar Lyapunov differential equations are also obtained. The established method includes some existing ones as special cases. Examples demonstrate the effectiveness of the proposed methods.