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Abstract

The Lyapunov inequality is an indispensable tool for stability analysis in linear control theory. It provides a necessary and sufficient condition for the stability of an autonomous linear-time invariant system in terms of the existence of a symmetric positive-definite Lyapunov matrix. This work proposes a new variant of this inequality in which the constituent Lyapunov matrix is allowed to be asymmetric. After analysing the properties of the proposed inequality for a class of matrices, we derive new results for the stabilisation of linear systems. Subsequently, we utilize the developed results to obtain sufficient conditions for the suboptimal linear quadratic control design problem, where addition to having an asymmetric Lyapunov matrix, which serves as a design matrix for this problem, we provide a characterization of the cost associated with the computed stabilizing suboptimal control laws by deriving an expression for the upper bound on cost in terms of the initial conditions of the system. We demonstrate the applicability of the proposed results using two numerical examples- one for suboptimal control design for a linear time-invariant system and another for the consensus (state-agreement) protocol design for a multi-agent system where-in we see how the asymmetry of the design matrix emerges as an inherent requirement for the problem.