## Abstract

In this article we establish new second order necessary and sufficient optimality conditions for a class of control-affine problems with a scalar control and a scalar state constraint. These optimality conditions extend to the constrained state framework the Goh transform, which is the classical tool for obtaining an extension of the Legendre condition.

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## Notes

- 1.
Actually \(H_u\) is continuous on

*B*since*p*does not jump on*B*.

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## Acknowledgments

We wish to thank the anonymous referees for their bibliographical advices. This work was partially supported by the European Union under the 7th Framework Pro-gramme FP7-PEOPLE-2010-ITN Grant Agreement Number 264735-SADCO. The last stage of this research took place while the first author was holding a postdoctoral position at IMPA, Rio de Janeiro, with CAPES-Brazil funding.

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## Appendix: On second order necessary conditions

### Appendix: On second order necessary conditions

### General constraints

We will study an abstract optimization problem of the form

where *X*, \(Y_E\), \(Y_I\) are Banach spaces, \(f:X\rightarrow {\mathbb {R}}\), \(G_E:X\rightarrow Y_E\), and \(G_I:X\rightarrow Y_I\) are functions of class \(C^2\), and \(K_I\) is a closed convex subset of \(Y_I\) with nonempty interior. The subindex *E* is used to refer to ‘equalities’ and *I* to ‘inequalities’.

Setting

we can rewrite problem (5.1) in the more compact form

We use \(F(P_A)\) to denote the set of feasible solutions of \((P_A)\).

###
*Remark 9*

We refer to [10] for a systematic study of problem \((P_A)\). Here we will take advantage of the product structure (that one can find in essentially all practical applications) to introduce a *non qualified version* of second order necessary conditions specialized to the case of quasi radial directions, that extends in some sense [10, Theorem3.50]. See Kawasaki [25] for non radial directions.

The *tangent cone* (in the sense of convex analysis) to \(K_I\) at \(y\in K_I\) is defined as

and the *normal cone* to \(K_I\) at \(y\in K_I\) is

In what follows, we shall study a nominal feasible solution \(\hat{x}\in F(P_A)\) that may satisfy or not the *qualification condition*

The latter condition coincides with the qualification condition in (2.21) which was introduced for the optimal control problem (P).

###
*Remark 10*

Condition (5.5) is equivalent to the Robinson qualification condition in [39]. See the discussion in [10, Section 2.3.4].

The *Lagrangian function* of problem \((P_A)\) is defined as

where we set \(\lambda :=(\beta ,\lambda _E,\lambda _I) \in {\mathbb {R}}_+ \times Y_E^* \times Y_I^*.\) Define the *set of Lagrange multipliers* associated with \( x \in F(P_A)\) as

Let \(y_I \in \mathop {\mathrm{int}}(K_I),\) \(y_I \ne G_I(\hat{x}).\) We consider the following auxiliary problem, where \((x,\gamma ) \in X \times {\mathbb {R}}\):

Note that we recognize the idea of a *gauge function* (see e.g. [40]) in the last constraint.

###
**Lemma 4**

Assume that \(\hat{x}\) is a local solution of \((P_A)\). Then \((\hat{x},0)\) is a local solution of \((AP_A)\).

###
*Proof*

We easily check that \((\hat{x},0)\in F(AP_A)\). Now take \((x,\gamma )\in F(AP_A).\) Let us prove that if \(-1/2 \le \gamma <0,\) then *x* cannot be closed to \(\hat{x}\) (in the norm of the Banach space *X*). Assuming that \(-1/2 \le \gamma <0,\) we get \(G_E(x)=0,\)
\(G_I(x) \in K_I + (-\gamma ) y_I \subseteq K_I\), and \(f(x) < f(\hat{x})\). Since \(\hat{x}\) is a local solution of \((P_A)\), the *x* cannot be too closed to \(\hat{x}.\) The conclusion follows.

The Lagrangian function of \((AP_A)\), in qualified form, is

or equivalently

Setting \(\hat{\lambda }=(\beta _0,\beta ,\lambda _E,\lambda _I)\), we see that the set of Lagrange multipliers of the auxiliary problem \((AP_A)\) at \((\hat{x},0)\) is

###
**Proposition 8**

Suppose that (5.5)(i) holds. Then, the mapping

is a bijection between \({\varLambda }(\hat{x})\) and \(\hat{{\varLambda }}_1\) (recall the definition in (2.16)).

###
*Proof*

Since (5.5)(i) holds, then we necessarily have that \((\beta ,\lambda _I) \ne 0\) for all \(\lambda =(\beta ,\lambda _E,\lambda _I) \in {\varLambda }(\hat{x}).\) Therefore, if \(\lambda _I = 0\) then \(\beta >0\) and \(\beta + \langle \lambda _I, G_I(x) - y_I \rangle > 0.\) If by the contrary, \(\lambda _I \ne 0,\) then \(\langle \lambda _I, G_I(x) - y_I \rangle > 0\) and again, \(\beta + \langle \lambda _I, G_I(x) - y_I \rangle > 0.\) Hence, the mapping in (5.11) is well-defined and is a bijection from \({\varLambda }(\hat{x})\) to \(\hat{{\varLambda }}_1,\) as we wanted to show.

###
**Theorem 6**

Let \(\hat{x}\) be a local solution of \((P_A)\), such that \(DG_E(\hat{x})\) is surjective. Then \(\hat{{\varLambda }}_1\) is non empty and bounded.

###
*Proof*

By lemma 4, \((\hat{x},0)\) is a local solution of \((AP_A)\). In addition the qualification condition for the latter problem at the point \((\hat{x},0)\) states as follows: there exists \((z,\delta ) \in \mathop {\mathrm{Ker}}DG_E(\hat{x}) \times {\mathbb {R}}\) such that

These conditions trivially hold for \((z,\delta )=(0,1).\) Hence, in view of classical results by e.g. Robinson [39], the conclusion follows.

### Second order necessary optimality conditions

Let us introduce the notation [*a*, *b*] to refer to the segment \(\{\rho a+(1-\rho )b;\ \text {for }\rho \in [0,1]\},\) defined for any pair of points *a*, *b* in an arbitrary vector space *Z*.

###
**Definition 6**

Let \(y\in K\). We say that \(z\in Y\) is a *radial direction* to *K* at *y* if \([y,y+\varepsilon z] \subset K\) for some \(\varepsilon >0\), and a *quasi-radial direction* if \(\mathop {\mathrm{dist}}(y+\sigma z,K) = o(\sigma ^2)\) for \(\sigma >0\).

Note that any radial direction is also quasi-radial, and both radial and quasi radial directions are tangent. With \(\hat{x}\in F(P_A)\), we associate the *critical cone*

###
**Definition 7**

We say that \(z\in C(\hat{x})\) is a *radial (quasi radial) critical direction* for problem \((P_A)\) if \(D G_I(\hat{x}) z\) is a radial (quasi radial) direction to \(K_I\) at \(G_I(\hat{x})\). We write \(C_{QR}(\hat{x})\) for the *set of quasi radial critical directions.* The critical cone \(C(\hat{x})\) is *quasi radial* if \(C_{QR}(\hat{x})\) is a dense subset of \(C(\hat{x})\).

It is immediate to check that \(C_{QR}(\hat{x})\) is a convex cone.

We next state *primal second order necessary conditions* for the problem \((P_A)\). Consider the following optimization problem, where \(z\in X\), \(w\in X\) and \(\theta \in {\mathbb {R}}\):

###
**Theorem 7**

Let \((\hat{x},0)\) be a local solution of \((AP_A)\), such that \(DG_E(\hat{x})\) is surjective, and let \(h\in C_{QR}(\hat{x})\). Then problem \((Q_z)\) is feasible, and has a nonnegative value.

###
*Proof*

We shall first show that \((Q_z)\) is feasible. Since \(DG_E(\hat{x})\) is surjective, there exists \(w \in X\) such that \(D G_E(\hat{x}) w + D^2 G_E(\hat{x})(z,z) = 0\). Since \(T_K(G_I(\hat{x}))\) is a cone, the last equation divided by \(\theta >0\) is equivalent to

Since \(y_I\in \mathop {\mathrm{int}}(K_I)\), we have that \(y_I - G(\hat{x})\in \mathop {\mathrm{int}}T_K(G_I(\hat{x}))\), and therefore the last constraint of \((Q_z)\) holds when \(\theta \) is large enough. So it does the first constraint, and hence, \((Q_z)\) is feasible.

We next have to show that we cannot have \((w,\theta _0)\in F(Q_z)\) with \(\theta _0<0.\) Let us suppose, on the contrary, that there is such a feasible solution \((w,\theta _0).\) Set \(\theta :=\frac{1}{2}\theta _0\). Then \(Df(\hat{x}) w + D^2f(\hat{x})(z,z) < \theta .\) Using (5.13) and \(y_I\in \mathop {\mathrm{int}}(K_i)\), we can easily show that, for some \(\varepsilon >0\):

Consider, for \(\sigma >0\), the path

By a second order Taylor expansion we obtain that \(G_E(x_\sigma ) = o(\sigma ^2)\). Since \(DG_E(\hat{x})\) is onto, by Lyusternik’s theorem [27], there exists a path \(x'_\sigma = x_\sigma + o(\sigma ^2)\), such that \(G_E(x'_\sigma )=0\). Assuming, without loss of generality, that \(G_I(\hat{x})=0\), we get

Setting

we can rewrite (5.16) as

Since *z* is a quasi radial critical direction, there exists \(k'_1(\sigma )\in K_I\) such that

and so,

Using (5.14) and \(G_I(\hat{x})=0\) we obtain

Therefore, for \(\sigma >0\) small enough

where we have used the fact that since \(0=G_I(\bar{x})\in K_I\), we have that (remember that \(\theta <0\)): \(\frac{1}{2}\sigma ^2 (-\theta ) (y_I + \varepsilon B) \subset K_I\).

We check easily that \(f(x'_\sigma ) <0\), and so, we have constructed a feasible path for \((AP_A)\), contradicting the local optimality of \((\hat{x},0)\).

We conclude that such a solution \((w,\theta _0)\) of \((Q_z)\) with \(\theta _0<0\) cannot exist and, therefore, \((Q_z)\) has nonnegative value.

We now present dual second order necessary conditions.

###
**Theorem 8**

Let \(\hat{x}\) be a local minimum of \((P_A)\), that satisfies the qualification condition (5.5). Then, for every \(z\in C_{QR}(\hat{x}),\)

###
*Proof*

Since problem \((Q_z)\) is qualified with a finite nonnegative value, by the convex duality theory [14], its dual has a nonnegative value and a nonempty set of solutions. The Lagrangian of problem \((Q_z)\) in qualified form (\(\beta _0=1\)) can be written as

where \(\lambda =(\beta ,\lambda _E,\lambda _I)\) as before, and so, the dual problem of \((Q_z)\) can be written as

The conclusion follows.

###
*Remark 11*

Whereas the above theorem follows from Cominetti [12] or Kawasaki [25], our proof avoids the concepts of second order tangent set and its associated calculus, used in these references. This considerably simplifies the proof.

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Aronna, M.S., Bonnans, J.F. & Goh, B.S. Second order analysis of control-affine problems with scalar state constraint.
*Math. Program.* **160, **115–147 (2016). https://doi.org/10.1007/s10107-015-0976-0

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### Mathematics Subject Classification

- 49K15
- 49K27