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
Actually \(H_u\) is continuous on B since p does not jump on B.
References
Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)
Aftalion, A., Bonnans, J.: Optimization of running strategies based on anaerobic energy and variations of velocity. SIAM J. Appl. Math. 74(5), 1615–1636 (2014)
Agrachev, A., Sachkov, Y.: Control theory from the geometric viewpoint. In: Encyclopaedia of Mathematical Sciences, 87, Control Theory and Optimization, II. Springer, Berlin (2004)
Agrachev, A.A., Stefani, G., Zezza, P.L.: Strong optimality for a bang–bang trajectory. SIAM J. Control Optim. 41, 991–1014 (2002)
Aronna, M.S.: Singular Solutions in Optimal Control: Second Order Conditions and a Shooting Algorithm. In: Technical report, ArXiv (2013). (Published online as arXiv:1210.7425, submitted)
Aronna, M.S., Bonnans, J.F., Dmitruk, A.V., Lotito, P.A.: Quadratic order conditions for bang-singular extremals. Numer. Algebra Control Optim. 2(3), 511–546 (2012)
Arutyunov, A.V.: On necessary conditions for optimality in a problem with phase constraints. Dokl. Akad. Nauk SSSR 280(5), 1033–1037 (1985)
Arutyunov, A.V.: Optimality Conditions: Abnormal and degenerate problems, Volume 526 of Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht (2000). Translated from the Russian by S. A. Vakhrameev
Bonnans, J.F., Hermant, A.: Revisiting the analysis of optimal control problems with several state constraints. Control Cybern. 38(4A), 1021–1052 (2009)
Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)
Bonnard, B., Faubourg, L., Launay, G., Trélat, E.: Optimal control with state constraints and the space shuttle re-entry problem. J. Dyn. Control Syst. 9(2), 155–199 (2003)
Cominetti, R.: Metric regularity, tangent sets and second order optimality conditions. J. Appl. Math. Optim. 21, 265–287 (1990)
de Pinho, M.R., Ferreira, M.M., Ledzewicz, U., Schaettler, H.: A model for cancer chemotherapy with state–space constraints. Nonlinear Anal. Theory Methods Appl. 63(5), e2591–e2602 (2005)
Ekeland, I., Temam, R.: Convex Analysis and Variational Problems, Volume 1 of Studies in Mathematics and its Applications. North-Holland, Amsterdam, 1976. French edition: Analyse convexe et problèmes variationnels. Dunod, Paris (1974)
Felgenhauer, U.: Optimality and sensitivity for semilinear bang–bang type optimal control problems. Int. J. Appl. Math. Comput. Sci. 14(4), 447–454 (2004)
Frankowska, H., Tonon, D.: Pointwise second-order necessary optimality conditions for the Mayer problem with control constraints. SIAM J. Control Optim. 51(5), 3814–3843 (2013)
Gabasov, R., Kirillova, F.M.: High-order necessary conditions for optimality. J. SIAM Control 10, 127–168 (1972)
Goh, B.S.: Necessary conditions for singular extremals involving multiple control variables. J. SIAM Control 4, 716–731 (1966)
Goh, B.S., Leitmann, G., Vincent, T.L.: Optimal control of a prey–predator system. Math. Biosci. 19, 263–286 (1974)
Graichen, K., Petit, N.: Solving the Goddard problem with thrust and dynamic pressure constraints using saturation functions. In: 17th World Congress of The International Federation of Automatic Control, Volume Proceedings of the 2008 IFAC World Congress, pp. 14301–14306, Seoul (2008). IFAC
Hestenes, M.R.: Applications of the theory of quadratic forms in Hilbert space to the calculus of variations. Pac. J. Math. 1(4), 525–581 (1951)
Hoffman, A.: On approximate solutions of systems of linear inequalities. J. Res. Natl. Bur. Stand Sect. B Math. Sci. 49, 263–265 (1952)
Jacobson, D.H., Speyer, J.L.: Necessary and sufficient conditions for optimality for singular control problems: a limit approach. J. Math. Anal. Appl. 34, 239–266 (1971)
Karamzin, D.Y.: Necessary conditions for an extremum in a control problem with phase constraints. Zh. Vychisl. Mat. Mat. Fiz. 47(7), 1123–1150 (2007)
Kawasaki, H.: An envelope-like effect of infinitely many inequality constraints on second order necessary conditions for minimization problems. Math. Program. 41, 73–96 (1988)
Kelley, H.J.: A second variation test for singular extremals. AIAA J. 2, 1380–1382 (1964)
Lyusternik, L.: Conditional extrema of functions. Math. USSR-Sb 41, 390–440 (1934)
Malanowski, K., Maurer, H.: Sensitivity analysis for parametric control problems with control-state constraints. Comput. Optim. Appl. 5(3), 253–283 (1996)
Maurer, H.: On optimal control problems with bounded state variables and control appearing linearly. SIAM J. Control Optim. 15(3), 345–362 (1977)
Maurer, H., Kim, J.-H.R., Vossen, G.: On a state-constrained control problem in optimal production and maintenance. In: Deissenberg, C., Hartl, R.F. (eds.) Optimal Control and Dynamic Games, Applications in Finance, Management Science and Economics, vol. 7, pp. 289–308. Springer, Berlin (2005)
Maurer, H., Osmolovskii, N.P.: Second order optimality conditions for bang–bang control problems. Control Cybern. 32, 555–584 (2003)
McDanell, J.P., Powers, W.F.: Necessary conditions joining optimal singular and nonsingular subarcs. SIAM J. Control 9, 161–173 (1971)
Milyutin, A.A., Osmolovskii, N.N.: Calculus of Variations and Optimal Control. American Mathematical Society, Providence (1998)
Osmolovskii, N.P., Maurer, H.: Equivalence of second order optimality conditions for bang-bang control problems. I. Main results. Control Cybern. 34(3), 927–950 (2005)
Osmolovskii, N.P., Maurer, H.: Applications to Regular and Bang–Bang Control, Volume 24 of Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2012). Second-order necessary and sufficient optimality conditions in calculus of variations and optimal control
Poggiolini, L., Spadini, M.: Strong local optimality for a bang–bang trajectory in a Mayer problem. SIAM J. Control Optim. 49, 140–161 (2011)
Poggiolini, L., Stefani, G.: Bang-singular-bang extremals: sufficient optimality conditions. J. Dyn. Control Syst. 17(4), 469–514 (2011)
Rampazzo, F., Vinter, R.: Degenerate optimal control problems with state constraints. SIAM J. Control Optim. 39(4), 989–1007 (2000). (electronic)
Robinson, S.M.: First order conditions for general nonlinear optimization. SIAM J. Appl. Math. 30, 597–607 (1976)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Russak, I.B.: Second-order necessary conditions for general problems with state inequality constraints. J. Optim. Theory Appl. 17(112), 43–92 (1975)
Russak, I.B.: Second order necessary conditions for problems with state inequality constraints. SIAM J. Control 13, 372–388 (1975)
Schattler, H.: A local field of extremals near boundary arc-interior arc junctions. In: 44th IEEE Conference on Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC ’05. pp. 945–950 (2005)
Schättler, H.: Local fields of extremals for optimal control problems with state constraints of relative degree 1. J. Dyn. Control Syst. 12(4), 563–599 (2006)
Seywald, H., Cliff, E.M.: Goddard problem in presence of a dynamic pressure limit. J. Guid. Control Dyn. 16(4), 776–781 (1993)
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
1.1 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.
1.2 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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DOI: https://doi.org/10.1007/s10107-015-0976-0