How can I solve a problem in the calculus of variations when the Lagrangian does not depend on the derivative and there is a salvage function, i.e. $J[x]=\int_{0}^{T} F(x, t) \, dt + B[x(T)]$ ? I can't obviously apply the usual condition $\left. \frac{\partial F(t, x, x')}{\partial x'}\right|_{t=T} =B[x(T)]$
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Without controlling the derivative, the problem is a bit ill posed, as $x(T)$ can take any value independent of what $x$ does in the rest of the integral. So your solution will be a minimizer of $\int_0^T F(x,t)dt$, with simply $x(T)$ changed to a global minimum of $B$, as this does not affect the integral.
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$\begingroup$ Many thanks. However, wouldn't this approach lead to a discontinuity of x(t) for t=T? Is there any way to enforce continuity? Indeed, the problem corresponds to a reasonable physical model. $\endgroup$– AnalyticCommented Aug 12 at 20:27
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$\begingroup$ You cannot enforce continuity without controlling the derivative. You can approximate that discontinuous solution by continuous solutions deviating from the optimum only in $(T-\epsilon,T)$, which for $\epsilon \to 0$ will converge pointwise and in value to that jump. So assuming the minima of $F(.,T)$ and $B$ do not accidentally sync up, the problem has no continuous minimizer. $\endgroup$– mlkCommented Aug 13 at 5:49