Singular Trajectories in Optimal Control
Abstract
Singular trajectories arise in optimal control as singularities of the endpoint mapping. Their importance has long been recognized, at first in the Lagrange problem in the calculus of variations where they are lifted into abnormal extremals. Singular trajectories are candidates as minimizers for the timeoptimal control problem, and they are parameterized by the maximum principle via a pseudoHamiltonian function. Moreover, besides their importance in optimal control theory, these trajectories play an important role in the classification of systems for the action of the feedback group.
Keywords
Endpoint mapping Abnormal extremals PseudoHamiltonian Saturation problem in NMR Martinet flat case in SR geometryIntroduction
The concept of singular trajectories in optimal control corresponds to abnormal extrema in optimization. Suppose that a point \({x}^{{\ast}}\in X \sim eq{\mathbb{R}}^{n}\) is a point of extremum for a smooth function \(\mathcal{L} : {\mathbb{R}}^{n} \rightarrow\mathbb{R}\) under the equality constraints F(x) = 0 where F : X → Y is a smooth mapping into \(Y \sim eq{\mathbb{R}}^{p}\), p < n. The Lagrange multiplier rule (Agrachev et al. 1997) asserts the existence of nonzero pairs \((\lambda _{0}{,\lambda }^{{\ast}})\) of Lagrange multipliers such that \(\lambda _{0}\mathcal{L}'({x}^{{\ast}}) {+\lambda }^{{\ast}}F'({x}^{{\ast}}) = 0\). The normality condition is given by λ _{0}≠0, and the abnormal case corresponds to the situation when the rank of F′(x ^{∗}) is strictly less than p.
Definition
Consider a system of \({\mathbb{R}}^{n}\): \(\frac{dx} {dt} (t) = F(x(t),u(t))\) where F is a smooth mapping from \({\mathbb{R}}^{n} \times{\mathbb{R}}^{m}\) into \({\mathbb{R}}^{n}\). Fix \(x_{0} \in{\mathbb{R}}^{n}\) and T > 0. The endpoint mapping is the mapping \({E}^{x_{0},T} : u(.) \in \mathcal{U}\rightarrow x(T,x_{0},u)\) where \(\mathcal{U}\subset{L}^{\infty }[0,T]\) is the set of admissible controls such that the corresponding trajectory x(., x _{0}, u) is defined on [0, T]. A control u(. ) and its corresponding trajectory are called singular on \([0,T]\) if \(u(.) \in \mathcal{U}\) is such that the Fréchet derivative \({E'}^{x_{0},T}\) of the endpoint mapping is not of full rank n at u(. ).
Fréchet Derivative and Linearized System
Proposition 1.
Computation of the Singular Trajectories and Pontryagin Maximum Principle
According to the previous computations, a control u(. ) with corresponding trajectory x(. ) is singular on [0, T] if the Fréchet derivative \({E'}^{x_{0},T}\) is not of full rank at u(. ). This is equivalent to the condition that the linearized system is not controllable (Lee and Markus 1967).
Such a condition is difficult to verify directly since the linearized system is timedepending and the computation is associated to the Maximum Principle (Pontryagin et al. 1962).
Proposition 2.
Application to the Lagrange Problem
The Role of Singular Extremals in Optimal Control
While the traditional treatment in optimization of singular extremals is to consider them as a pathology, in modern optimal control, they play an important role which is illustrated by two examples from geometric optimal control.
Singular Trajectories in Quantum Control

The vertical line y = 0, corresponding to the zaxis of revolution

The horizontal line \(z = \frac{\gamma } {2(\gamma \Gamma )}\)
The interesting physical case is when 2Γ > 3γ where the vertical singular line is such that \(1 < \frac{\gamma } {2(\gamma \Gamma )} < 0\). In this case, the time minimum solution is represented on Fig. 1. On Fig. 2 we draw the experimental solution in the deoxygenated blood case, compared with the standard inversion recovery sequence.
Abnormal Extremals in SR Geometry
Summary and Future Directions
Singular trajectories play an important role in many optimal control problem such as in quantum control and cancer therapy (Schättler and Ledzewicz 2012). They have to be carefully analyzed in any applications; in particular in Boscain and Piccoli (2006) the authors provide for singleinput systems in two dimensions a classification of optimal synthesis with singular arcs.
Additionally, from a theoretical point of view, singular trajectories can be used to compute feedback invariants for nonlinear systems (Bonnard and Chyba 2003). In relation, a purely mathematical problem is the classification of distributions describing the nonholonomic constraints in subRiemannian geometry (Montgomery 2002).
CrossReferences
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