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Optimal control of ODE models
Published in Alfio Borzì, Modelling with Ordinary Differential Equations, 2020
Next, we discuss an example that illustrates the notion of singular controls. A singular control is one for which the Legendre-Clebsch (LC) condition is not satisfied with strict inequality anywhere along the extremal. Thus, we do not have any concavity property of the HP function. Equivalently, we can say that u is singular if ∂2H∂u2(x,y,u,p)=0 along the optimal trajectory. In particular, if H is linear in one (or more) components of the control function, then the extremal is singular. In this case, a generalised form of the LC necessary conditions is required that provides a characterisation of a singular extremal; see [14].
A two-step method for solving singular control problems
Published in International Journal of Control, 2023
Singular optimal control problems are difficult to solve, since the gradient of the Hamiltonian does not provide any information with respect to the control, when it is zero. For simpler problems, the control for the singular arc can be explicitly determined by imposing higher-order derivative conditions on the switching function, Equation (6), for the singular sections (Betts, 2010) for some . The largest number of differentiation steps, κ, in Equation (8), required to obtain an explicit expression for control is called the order of the singular control problem, . If, for some , does not appear explicitly in Equation (8) for any κ, then the order of the singular control problem is defined as infinite. For finite order singular control problems, the order always turns out to be even and it is necessary that the generalised Legendre-Clebsch condition be satisfied for an optimal solution over the singular arc (Krener, 1977).
Observer path planning for maximum information
Published in Optimization, 2022
Figure 4 conveys further information: The optimal control switches from the bang arc deg/s to singular control, which is an interior arc, i.e. the constraint on the curvature is inactive, as also described in Corollary 2.1. The numerical experiments conducted so far suggest that the observer trajectory reaches the target position with infinite information when .