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Convex analysis in the calculus of variations. (English) Zbl 1011.49013
Hadjisavvas, Nicolas (ed.) et al., Advances in convex analysis and global optimization. Honoring the memory of C. Caratheodory (1873-1950). Dordrecht: Kluwer Academic Publishers. Nonconvex Optim. Appl. 54, 135-151 (2001).
Let the Lagrange problem \(({\mathcal P}_0)\) be given by \[ \text{minimize }J_0(x):= \int^{\tau_1}_{\tau_0} L(t, x(t),\dot x(t)) dt, \]
\[ \text{subject to }x(\tau_0)= \xi_0,\quad x(\tau_1)= \xi_1, \] 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 \(({\mathcal P}_0)\).
As 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 \({\mathcal A}^\infty_n[\tau_0, \tau_1]\) \[ \begin{alignedat}{2} &y= \nabla_v L(t,x,\dot x),\;\dot y= \nabla_x L(t,x,\dot x)\quad &&\text{(Euler/Lagrange)}\\ & L(t,x,v)\geq L(t,x,\dot x)+ \langle\nabla_v L(t,x,\dot x), v-\dot x\rangle \forall v\quad &&\text{(Weierstraß)}\\ & \nabla^2_{vv} L(t,x,\dot x)\text{ is positive semi-definite}\quad &&\text{(Legendre)}\\ &\dot x= \nabla_y H(t,x,y),\;\dot y=-\nabla_x H(t,x,y)\quad &&\text{(Hamilton)}\end{alignedat} \] (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 \[ (\dot y,y)\in \partial_{x,v}L(t,x,\dot x), \]
\[ \dot x\in \partial_y H(t,x,y),\quad \dot y= \partial_x(- H(t,x,y)). \] In this fully convex case, the necessary optimality conditions are also sufficient for the optimality of a feasible arc \(x(t)\).
In the second part of the paper, the author extends his results to the generalized Bolza problem \(({\mathcal P})\) according to \[ \text{minimize }J(x):= \int^{\tau_1}_{\tau_0} L(t, x(t),\dot x(t)) dt+ l(x(\tau_0), x(\tau_1)) \] with free endpoints and a convex cost term \(\ell:\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}\).
For the entire collection see [Zbl 0968.00020].

49J52 Nonsmooth analysis
26B25 Convexity of real functions of several variables, generalizations
49J10 Existence theories for free problems in two or more independent variables
49K10 Optimality conditions for free problems in two or more independent variables