Symplectic difference systems: variable stepsize discretization and discrete quadratic functionals

Discrete quadratic functionals with variable endpoints for variable stepsize symplectic difference systems are considered. A comprehensive study is presented for characterizing the positivity of such functionals in terms of conjugate intervals, conjoined bases, and implicit and explicit Riccati equations with various forms of boundary conditions. Moreover, necessary conditions for the nonnegativity of these functionals are obtained in terms of the above notions. Furthermore, we show that a variable stepsize discretization of a continuous-time nonlinear control problem leads to a discrete linear quadratic problem and a Hamiltonian difference system, which are special cases of their symplectic counterparts.

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