Dissipativity for dual linear differential inclusions through conjugate storage functions

Tools from convex analysis are used to show how dissipativity properties, expressed in terms of convex storage functions, translate when passing from a linear differential inclusion (LDI) to its dual. As special cases, it is shown that a convex, positive definite function is a Lyapunov function for an LDI if and only if its convex conjugate is a Lyapunov function for the LDI's dual, and that passivity and finite L2-gain are preserved when passing from an LDI with input and output to its dual. Also established is the duality between stabilizability and detectability, including stabilizable and detectable dissipativity, for dual LDIs. Finally, with examples we show how duality effectively doubles the number of tools available for assessing stability and performance of LDIs.