Symplectic difference systems: Variable stepsize discretization and discrete quadratic functionals
The authors consider the positivity and the nonnegativity of some discrete quadratic functionals with variable endpoints for variable stepsize symplectic difference systems. Seventeen theorems are given for the above-mentioned properties of such functionals in terms of conjugate intervals, conjoined bases, and implicit and explicit Riccati equations with various forms of boundary conditions. Moreover, they show that a variable stepsize discretization of a continuous time nonlinear control problem of the form \[ \begin{cases} \text{minimize} \quad & G(x,u):=K\bigl(x(a), x(b)\bigr)+ \int^b_a g\bigl(t,x(t), u(t)\bigr) dt,\\ & x'(t)=f\bigl(t,x(t), u(t)\bigr),\;\psi\bigl( t,u(t) \bigr)=0;\;t\in[a,b],\\ & \varphi\bigl(x(a), x(b)\bigr)= 0\end{cases} \] leads to a discrete linear quadratic problem and a Hamiltonian difference system, which are special cases of their symplectic counterparts. A number of interesting remarks, lemmas and corollaries are also indicated herein.
scientific article; zbMATH DE number 1917805
scientific article