Invariants in Optimal Control: An Exact Solution of the Optimal Stabilization Problem

The stabilizing optimal feedback is a function on the separatrix of stable points of the associ­ ated Hamiltonian system. Three geometric objects — the symplectic form, Hamiltonian vector field, and Lyapunov function, generating the separatrix — are invariantly attached to the optimal control system under canonical transformations of the phase space. The separatrix equations can be written in terms of these invariants through invariant operations. There is a computable representative of the equivalence class, containing the original system. It is its linear approximation system at the stable point. Subject Classification Numbers: 49J15, 49J21, 37J05, 37J15. Introduction The problem of optimal stabilization is normally reformulated in terms of the associated Hamilto­ nian system. The separatrix of stable points with respect to the origin of the Hamiltonian system is a Lagrangian manifold of the maximal dimension, which consists of all integral curves going to the origin from a maximal neighbourhood. The standard procedure to compute a Lagrangian manifold is the Cauchy characteristics method, extending an initial Lagrangian surface transversal to the integral curves of the Hamiltonian vector field along them to a Lagrangian surface of the dimension larger by one. In the optimal stabilization problem there is no known tranversal surface of this type. The Cauchy charac­ teristics method fails due to the incorrect initial conditions — only one point, the origin of the phase space, is known. In such cases, a powerful method of invariants associated to the object can be successfully used. Invariants are just functions constructed from the base intrinsic objects using operations, invariant under admissible maps, such as the canonical transformations in Hamiltonian Mechanics. Having invariants on the orbit of the transformation group and the values of them for a canonical representative of the orbit, one automatically has these invariants and their values, distributed over all the orbit points. The paper proposes to use the following invariants — the symplectic form, Hamiltonian vector field, separatrix of stable points, generating Lyapunov function of the separatrix — to obtain exact formulas of the separatrix and optimal feedback. A canonical representative whose invariants are com­ putable is the linear approximation of the system at the stable point. The statement and analysis of the optimal stabilization problem from the point of HamiltonianMe­ chanics can be found in [1], the notions of Hamiltonian Mechanics in [2], vector fields and differential forms in [3], general ideas on local invariants of differential­geometric objects in [4, 5, 6, 7]. It should be noticed that the method of invariants is a powerful tool and undeservedly forgotten. The author used it in absolutely different areas such as Statistics and Data Analysis with the same ideology although with a different category­theoretic approach [8, 9] to get a high precision prediction tool and invariants of data. Bulletin of Mathematical Sciences and Applications Submitted: 2018-12-20 ISSN: 2278-9634, Vol. 21, pp 9-14 Revised: 2019-05-22 doi:10.18052/www.scipress.com/BMSA.21.9 Accepted: 2020-01-02 CC BY 4.0. Published by SciPress Ltd, Switzerland, 2019 Online: 2019-12-31 This paper is an open access paper published under the terms and conditions of the Creative Commons Attribution license (CC BY) (https://creativecommons.org/licenses/by/4.0) The Problem Formulation A smooth optimal stabilization problem is formulated as the following. For a control system ẋ = f(x, u), x ∈ R, u ∈ R, f(0, 0) = 0, (1)


Introduction
The problem of optimal stabilization is normally reformulated in terms of the associated Hamilto nian system.
The separatrix of stable points with respect to the origin of the Hamiltonian system is a Lagrangian manifold of the maximal dimension, which consists of all integral curves going to the origin from a maximal neighbourhood. The standard procedure to compute a Lagrangian manifold is the Cauchy characteristics method, extending an initial Lagrangian surface transversal to the integral curves of the Hamiltonian vector field along them to a Lagrangian surface of the dimension larger by one. In the optimal stabilization problem there is no known tranversal surface of this type. The Cauchy charac teristics method fails due to the incorrect initial conditions -only one point, the origin of the phase space, is known.
In such cases, a powerful method of invariants associated to the object can be successfully used. Invariants are just functions constructed from the base intrinsic objects using operations, invariant under admissible maps, such as the canonical transformations in Hamiltonian Mechanics.
Having invariants on the orbit of the transformation group and the values of them for a canonical representative of the orbit, one automatically has these invariants and their values, distributed over all the orbit points.
The paper proposes to use the following invariants -the symplectic form, Hamiltonian vector field, separatrix of stable points, generating Lyapunov function of the separatrix -to obtain exact formulas of the separatrix and optimal feedback. A canonical representative whose invariants are com putable is the linear approximation of the system at the stable point.
The statement and analysis of the optimal stabilization problem from the point of Hamiltonian Me chanics can be found in [1], the notions of Hamiltonian Mechanics in [2], vector fields and differential forms in [3], general ideas on local invariants of differentialgeometric objects in [4,5,6,7].
It should be noticed that the method of invariants is a powerful tool and undeservedly forgotten. The author used it in absolutely different areas such as Statistics and Data Analysis with the same ideology although with a different categorytheoretic approach [8,9] to get a high precision prediction tool and invariants of data.

The Problem Formulation
A smooth optimal stabilization problem is formulated as the following. For a control systeṁ and a functional where: f and ω are smooth functions, ω is nondegenerated at 0, that is its quadratic approximation at x = 0, u = 0 is represented by a matrix with a nonzero determinant, find a smooth stabilizing optimal feedback u = u opt (x) , that is, a smooth function, providing stability of the origin x = 0 of the control system 1 with u = u opt (x) and minimum of the functional 2 in the sense, that is required to stabilize the system 1 at the origin.
Both sides of inequality 3 are Lyapunov functions of the corresponding systems of ODEs. It is wellknown, that when the linear approximation system of the system 1 at x = 0, u = 0 is controllable, the problem 1, 2, 3 has a unique solution u opt : W → R m on a maximal neighbourhood W of the origin x = 0.
The function u opt (x), x ∈ W , is a fucntion obtained as where V x j = ∂V ∂x j , j = 1, . . . , n, are the first derivatives of the Lyapunov function V (x) as in 3 which satisties the HamiltonJacobi equation The function ϕ(x i , V j ) is the Hamiltonian of the problem. Coordinate functions x i , i = 1, . . . , n, and V j , j = 1, . . . , n, are respectively coordinates and impulses in the phase space R 2n .

Reformulation of the Problem in the Phase Space
There are three main geometric objects in the phase space associated to the problem 1, 2, 3. They are the symplectic 2form the Hamiltonian vector field 10 BMSA Volume 21 and the separatrix S + of stable points with respect to the origin x i = 0, V j = 0, i, j = 1, . . . , n, of the Hamiltonian vector field 7.
The field 7 is obtained from dϕ as a unique vector field such that the substitution brings i Idϕ Ω = dϕ.
The optimal stabilizing feedback u opt (x i , V j ) as in 4 is a vectorvalued function on S + : So, the problem is to calculate the separatrix S + . The object S + is a Lagrangian manifold, that is, one nullifying the symplectic form Ω S + = 0, with the generating Lyapunov function V (x). The manifold S + is given as Since, there is only one known point of the separatrix -the origin of the phase space R 2n -the method of Cauchy characteristics does not work.
The generating function of a Lagrangian manifold is determined uniquely up to a constant modulo the equations of the Lagrangian manifold, as in the following exact sequence The

Method of Invariants
The method is based on two assumptions. The first one is specific, that for any small ε neigh bourhood U ε of the orgin 0 ∈ R 2n there exists t > 0 such that the preimage e −t·Idϕ (U ε ) of the flow, generated by the field Idϕ, contains the intersection of the separatrix S + of stable points of the vector field Idϕ and an arbitrarily big chosen neighbourhood of the origin 0 ∈ R 2n . For any t ∈ R, this flow is a canonical transformation, preserving the symplectic form 6, vector field 7, and mapping the separatrix S + to its small part around 0 ∈ R 2n . It follows, that the triple (Ω, Idϕ, S + ) is isomorphic to the similar one (Ω, Idϕ l , S + l ), where Idϕ l is a linearization of the field Idϕ at point 0 ∈ R 2n , and S + l is a separatrix of stable points of the field Idϕ l . The second assumption is general. Any consequences of the triple, obtained by operations not alternating under admissible transformations, make intrinsic sense of the triple.
The vector field Idϕ has a stable point at the origin 0 ∈ R 2n . When this point is nondegenerated, it is hyperbolic one and two separatrices S + and S − of dimension n go through it, consisting respectively of the points approaching to the origin and going away of it. The phase portrait of the vector field Idϕ is equivalent to one of the linearization Idϕ l in a maximal neighbourhood of the origin 0 ∈ R 2n .
The field Idϕ l has two corresponding linear separatrices S + l and S − l of the form V i = where . . . , x n ) are nvectors, vanishes on the separatrices S + , S − , that is, Consequently, the matrix A satisfies to the algebraic quadratic Riccati equation For the stated problem equation 13 has two nondegenerated solutions -the positively and nega tively determined matrices A, corresponding respectively to S + l and S − l . They can be found arbitrarily precise. The generating function of It is convenient to take c = 0 for the canonical transformations, preserving the origin x = 0, and so,

An Example
For the control system and functional it is required to find the optimal stabilizing feedback u opt (x).
Applying the above technique we get the following.
is the optimal feedback on S + .
is the Hamiltonian.

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is the Hamiltonian vector field.
are respectively the quadratic approximation Hamiltonian and linearized Hamiltonian vector field.
is the generating function of S + l in the linearized problem. It can be checked for this particular case that where V (n) l is the nth derivative of function V l along the vector field Idϕ l . So, to get invariant equa tions of S + it is sufficient to compute just three derivatives.
one can obtain implicit equations of S + for the given problem Since the solution of equations 25 of the form V i = ∂ ∂x i V (x), i = 1, 2, is unique it is sufficient to know just one arbitrary point (x 1 * , x 2 * , V 1 * , V 2 * ) satisfying the equations, express after the differentials dV i , i = 1, 2, from the corresponding linear system, and approximately calculate the required surface.
For the given case an exact solution can be found with an appropriate substitution. The result is The optimal stabilizing feedback is where (x 1 , x 2 ) ∈ W ⊂ {(x 1 , x 2 ) | − π 2 < x 1 , x 2 < π 2 }, the set W is a maximal neighbourhood of 0 ∈ R 2 , where S + exists.

Conclusion
The method of invariants applied to the optimal stabilization problem reduces it to algebraic cal culations and creates exact implicit equations of the separatrix surface where the optimal feedback function is determined.
Although, in a general case the obtained equations are hardly computable exactly, they admit com putable linearization and computation by step of any point starting from one known.