On the strong local optimality for state-constrained control-affine problems

Abstract

In this article we establish first and second order sufficient optimality conditions for a class of single-input control-affine problems, in presence of a scalar state constraint. We consider strong-local optimality (that is, the $C^0$ topology in the state space). The minimum-time and the Mayer problem are addressed. We restrict our analysis to extremals containing a bang arc, a single boundary arc, followed by a finite number of bang arcs. The sufficient conditions are expressed as a strengthened version of the necessary ones, plus the coerciveness of a suitable finite-dimensional quadratic form. The sufficiency of the given conditions is proven via Hamiltonian methods.

Appendix A: On the computation of first and second variation

In this section, we provide more details on the constructions of the first and second variations of the functional J. As in Sect. 4, we focus on the minimum-time problem.

We start by recalling the coordinate expression of the reference extremal. Choosing any (local) coordinate chart on $T^*M$ and setting ${\widehat{\lambda }}(t) = \big ({\widehat{\mu }}(t), {\widehat{\xi }}(t)\big )$, Eq. (2.5c) implies that ${\widehat{\mu }}(t)$ is a solution to the following linear ODE

$$\begin{aligned} {\dot{\mu }}(t) = -\mu (t)\textrm{D}{{\widehat{f}}}_t({\widehat{\xi }}(t)) -{\widehat{\eta }}(t)\, \textrm{d}c({\widehat{\xi }}(t)). \end{aligned}$$

Then, for every $t \in [0,T]$, we have

$$\begin{aligned} {\widehat{\mu }}(t) = \Big ({\widehat{\mu }}({{\widehat{t}}}_2) - \int \nolimits _{{{\widehat{t}}}_2}^t {\widehat{\eta }}(s) \, \textrm{d}{{\widehat{c}}}_s({\widehat{x}}_{2}) \, \textrm{d}s \Big ) {\widehat{S}}^{-1}_{t \, *}, \end{aligned}$$

(A.1)

where we recall that $\widehat{\eta }$ is supported on $[{{\widehat{t}}}_1,{{\widehat{t}}}_2]$.

Let us now prove that the first variation, with respect to and u, of the cost (4.5) is null. Using (A.1), we can see that

$$\begin{aligned} \textrm{d}{\widehat{\alpha }}({\widehat{x}}_2)={\widehat{\ell }}_2+\int \nolimits _{{{\widehat{t}}}_1}^{{{\widehat{t}}}_2} {\widehat{\eta }}(s) \, \textrm{d}{{\widehat{c}}}_s({\widehat{x}}_{2}) \, \textrm{d}s, \qquad \textrm{d}{\widehat{\beta }}({\widehat{x}}_2)=-{\widehat{\ell }}_T{{\widehat{S}}}_{T*}=-{\widehat{\ell }}_2, \end{aligned}$$

Then we have that

Thanks to (4.3), the first variation of J with respect to u boils down to

which is null by PMP. Finally

where we recall that the $\delta _k$ are defined in Eq. (4.6). The first addendum is null because of the constraint (4.9), and the second addendum is null since $F_0({\widehat{\ell }}_k) = p_0$ for any k, so that

$$\begin{aligned} \sum _{k=3}^{N+1}\delta _k F_0({\widehat{\ell }}_k) = p_0 \sum _{k=3}^{N+1}\delta _k = 0. \end{aligned}$$

Differentiating twice the cost J with respect to and u, we obtain an expression which depends on $\delta u_0$ only by means of the quantities $\delta _3, \ldots , \delta _{N+1}$, i.e. we get

(A.2)

(A.3)

(A.4)

(A.5)

(A.6)

with $(\delta y,\delta u,\varvec{\delta })\in W$.

Exchanging the order of integration and relabeling the variables, we remark that

so (A.2) can be written as

Also, we have that

$$\begin{aligned} (\hbox {A.5})&=\int \nolimits _{{{\widehat{t}}}_1}^{{{\widehat{t}}}_2} \Bigg (\int \nolimits _{{{\widehat{t}}}_1}^r \Big ( \int \nolimits _s^{{{\widehat{t}}}_2} {\widehat{\eta }}(s) \delta u(r)\delta u(\tau ) L_{k_r}L_{k_{\tau }} {{\widehat{c}}}_s ({\widehat{x}}_2) \textrm{d}\tau \Big ) \textrm{d}s\Bigg ) \textrm{d}r\\&=\int \nolimits _{{{\widehat{t}}}_1}^{{{\widehat{t}}}_2} \Bigg ( \int \nolimits _{{{\widehat{t}}}_1}^{{{\widehat{t}}}_2} \Big ( \int \nolimits _{{{\widehat{t}}}_1}^{\tau } {\widehat{\eta }}(s) \delta u(r)\delta u(\tau ) L_{k_r}L_{k_{\tau }} {{\widehat{c}}}_s ({\widehat{x}}_2) \textrm{d}s \Big )d\tau \\&\quad - \int \nolimits _{r}^{{{\widehat{t}}}_2} \Big ( \int \nolimits _{r}^{\tau } {\widehat{\eta }}(s)\delta u(r)\delta u(\tau ) L_{k_r}L_{k_{\tau }} {{\widehat{c}}}_s ({\widehat{x}}_2) \textrm{d}s \Big )d\tau \Bigg )\textrm{d}r; \end{aligned}$$

noticing that , we obtain that

Analogous computations show that

Summing up the addenda, we obtain the expression (4.10).