Convex analysis in the calculus of variations (Q2768041)

Let the Lagrange problem \(({\mathcal P}_0)\) be given by NEWLINE\[NEWLINE\text{minimize }J_0(x):= \int^{\tau_1}_{\tau_0} L(t, x(t),\dot x(t)) dt,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\text{subject to }x(\tau_0)= \xi_0,\quad x(\tau_1)= \xi_1,NEWLINE\]NEWLINE where \(L: [\tau_0,\tau_1]\times \mathbb{R}^n\times \mathbb{R}^n\to\mathbb{R}\) is a smooth function and the minimization takes place in a space of arcs \(x: [\tau_0,\tau_1]\to \mathbb{R}^n\), e.g., in the space \({\mathcal C}^1_n[\tau_0, \tau_1]\) of all continuously differentiable arcs or -- more general -- in the space \({\mathcal A}^p_n[\tau_0, \tau_1]\) \((p\in [1,\infty])\) of all absolutely continuous arcs with \(\dot x\in{\mathcal L}^p_n[\tau_0, \tau_1]\) a.e. The author points out the importance of convexity properties of the Lagrange function \(L(t,x,v)\) in connection with existence assertions but also with the characterization of solutions of the problem \(({\mathcal P}_0)\).NEWLINENEWLINENEWLINEAs a first result it is shown that in the larger space \({\mathcal A}^1_n[\tau_0, \tau_1]\) the minimum of \(({\mathcal P}_0)\) is attained if the function \(L\) is convex with respect to the variable \(v\) and satisfies a suitable growth condition. Also the necessary optimality conditions in \({\mathcal C}^1_n[\tau_0, \tau_1]\) and in \({\mathcal A}^\infty_n[\tau_0, \tau_1]\) NEWLINE\[NEWLINE\begin{alignedat}{2} &y= \nabla_v L(t,x,\dot x),\;\dot y= \nabla_x L(t,x,\dot x)\quad &&\text{(Euler/Lagrange)}\\ & L(t,x,v)\geq L(t,x,\dot x)+ \langle\nabla_v L(t,x,\dot x), v-\dot x\rangle \forall v\quad &&\text{(Weierstraß)}\\ & \nabla^2_{vv} L(t,x,\dot x)\text{ is positive semi-definite}\quad &&\text{(Legendre)}\\ &\dot x= \nabla_y H(t,x,y),\;\dot y=-\nabla_x H(t,x,y)\quad &&\text{(Hamilton)}\end{alignedat}NEWLINE\]NEWLINE (using the Hamilton function \(H(t,x,y)= \langle y,v\rangle- L(t,x,v)\) where \(v\) is replaced by the solution mapping of the equation \(y= \nabla_v L(t,x,v)\), assuming the strong Legendre condition) are closely connected with convexity of \(L\) with respect to \(v\). Obviously \(H(t,x,.)\) turns out to be the Fenchel conjugate of \(L(t,x,.)\). Moreover, assuming convexity of \(L\) even with respect of \((x,v)\) then the Euler/Lagrange and Hamilton conditions can be extended to the larger space \({\mathcal A}^1_n[\tau_0,\tau_1]\) using the subdifferential mapping according to NEWLINE\[NEWLINE(\dot y,y)\in \partial_{x,v}L(t,x,\dot x),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\dot x\in \partial_y H(t,x,y),\quad \dot y= \partial_x(- H(t,x,y)).NEWLINE\]NEWLINE In this fully convex case, the necessary optimality conditions are also sufficient for the optimality of a feasible arc \(x(t)\).NEWLINENEWLINENEWLINEIn the second part of the paper, the author extends his results to the generalized Bolza problem \(({\mathcal P})\) according to NEWLINE\[NEWLINE\text{minimize }J(x):= \int^{\tau_1}_{\tau_0} L(t, x(t),\dot x(t)) dt+ l(x(\tau_0), x(\tau_1))NEWLINE\]NEWLINE with free endpoints and a convex cost term \(\ell:\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}\).NEWLINENEWLINEFor the entire collection see [Zbl 0968.00020].