Remarks on necessary conditions for minimizers of one-dimensional variational problems
Let \(L: [0,1]\times{\mathbb R}^n\times{\mathbb R}^n\to{\mathbb R}\), and \[ J(u)=\int_0^1L(t,u(t),\dot u(t))dt \] be a functional defined for \(u\colon[0,1]\to{\mathbb R}^n\). In the paper, under weak assumptions on \(L\), the authors consider a local notion of convexity of \(L\) at a point and prove that it is equivalent to the sequential weak*-\(W^{1,\infty}([0,1];{\mathbb R}^n)\) lower semicontinuity of \(J\). More precisely, they prove that convexity of \(L(t,u_0(t),\cdot)\) at \(\dot u_0(t)\) with respect to a ball centred at \(\dot u_0(t)\) with fixed radius \(\varepsilon\) for a.e. \(t\in[0,1]\) is necessary and sufficient for the weak*-\(W^{1,\infty}([0,1];{\mathbb R}^n)\) lower semicontinuity of \(J\) at \(u_0\). Since sequential lower semicontinuity holds at a minimizer of \(J\), the above result provides the validity of the Weierstrass condition. Moreover, a version of the Euler-Lagrange equation, which involves the subgradient of \(L\) at \(\dot u_0\) is also proved. Eventually, the results are compared with the ones achieved in the framework of optimal control approach.
scientific article; zbMATH DE number 1741839
scientific article