A shooting algorithm for optimal control problems with singular Arcs. (English) Zbl 1275.49045

The authors investigate a shooting algorithm for the optimal control problem \(\varphi_0(x_0,x_T) \to \min\), \(\dot{x}_t=\sum_{i=0}^m u_{i,t} f_i(x_t)\) a.e. on \([0,T]\), \(\eta_j(x_0,x_T)=0\), \(j=1,\dots,d\), where the final time \(T\) is fixed, \(f_i: R^n \to R^n\) for \(i=0, \dots,m\) and \(\eta_j:R^{2n} \to R\) for \(j=1,\dots,d\). The functions \(\varphi_0\), \(f_i\), and \(\eta_j\) have Lipschitz-continuous second derivatives. In general, the shooting system has more equations than unknowns, and the Gauss-Newton method is used to compute a zero of the shooting system. This shooting algorithm is locally quadratically convergent, if the derivative of the shooting function is one-to-one at the solution. The main result of the paper asserts that the latter holds whenever a sufficient condition for weak optimality is satisfied. Numerical tests that validate the proposed method are included.


49M15 Newton-type methods
49K40 Sensitivity, stability, well-posedness
49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)


minpack; Cotcot; Bocop; Shoot
Full Text: DOI arXiv


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