New second-order optimality conditions for variational problems with \(C^2\)-Hamiltonians (Q2753220)

The author studies the generalized Bolza problem, NEWLINE\[NEWLINE\text{minimize} J(x)=\ell (x(0),x(1))+\int_0^1 L(t,x(t),\dot x(t))dt,NEWLINE\]NEWLINE over all absolutely continuous functions \(x\in W^{1,1}[0,1]\) satisfying \(\;\phi(x(0),x(1))=0,\) where \(\;\ell :\mathbb R^n\times \mathbb R^n\to \mathbb R,\;L:[0,1]\times \mathbb R^n\times \mathbb R^n\to \mathbb R\cup \{+\infty\},\;\phi:[0,1]\times \mathbb R^n\times \mathbb R^n\to \mathbb R^r,\;r\leq 2n.\;\) The Hamiltonian corresponding to the problem is \(H(t,x,p):= \text{sup}\{\langle p,v\rangle - L(t,x,v):v\in \mathbb R^n\}.\) The problem is more general than it appears to be, due to the fact that \(L\) is allowed to take the value \(+\infty\). The problem is studied from the point of view of the Hamiltonian \(H\). Necessary and sufficient conditions are obtained, respectively, in terms of the accessory problem, the existence of conjoined basis, and the existence of a solution to a Riccati equation with boundary conditions. Strong \(L^\infty\) and \(W^{1,s}\)-weak local minimum are distinguished. A similar approach is applied to the more general problem, NEWLINE\[NEWLINE\text{minimize} J(x,u)=\ell (x(0),x(1))+\int_0^1 g(t,x(t),u(t))dt,NEWLINE\]NEWLINE subject to \(\dot x(t)=f(t,x(t),u(t))\;\text{a.e.},\;u(t)\in U\;\text{a.e.},\;\phi(x(0),x(1))=0,\) where \(f, g\) are \((\mathcal L\times \mathcal B)-\)measurable and continuous in \((x,u),\;U\) is closed in \(\mathbb R^m\) and \(\ell\) is lower semicontinuous. The associated Hamiltonian is \(H(t,x,p):= \text{sup}\{ p\cdot f(t,x,u) - g(t,x,v)\;|u\in U\}.\)