Multitarget Linear-Quadratic Control Problem : Semi-Infinite Interval

We consider multitarget linear-quadratic control problem on semi-infinite interval. We show that the problem can be reduced to a simple convex optimization problem on the simplex.


Introduction
Let (H, <, >) be a Hilbert space, Z be its closed vector subspace, h 1 , • • • , h m and c be vectors in H. Consider the following optimization problem: Here • is the norm in H induced by the scalar product <, >.In [FM], we analyzed (1) using duality theory for infinite-dimensional second-order cone programming.We obtained a reduction of this problem to a finite dimensional second-order cone programming and applied this result to a multi-target linearquadratic control problem on a finite time interval.In this paper, we consider a reduction (1) to even simpler optimization problem of minimization of convex quadratic function on the (m−1) dimensional simplex.We then apply this result to the analysis of a multi-target linear-quadratic control problem on semi-infinite time interval.We show that the coefficients of the quadratic function admit a simple expressions in term of the original data.
2 Reduction to a simple quadratic programming problem Consider the Lagrange function Notice that despite the fact that our original problem is infinite-dimensional, the usual KKT theorem holds true (see e.g.[MT],p.72 ).It is also clear that Slater conditions are satisfied.Hence, optimality condition for (2) -(4) take the form where Here π Z : H → Z is the orthogonal projection.Let us form the Lagrange dual of (2), (3),(4).Consider Using ( 7), (8), we obtain where Notice that for any Here π Z ⊥ : H → Z ⊥ is the orthogonal projection of H onto orthogonal complement Z ⊥ of Z.To further simplify (9), introduce the notation: Hence, according to ( 9) We, hence, arrive at the following expression of ϕ: We can simplify (11) somewhat.Notice that Consequently, Here, Hence, the Lagrange dual to (2), ( 3), (4) takes the form: ) is an optimal solution to ( 13), ( 14), we can recover the optimal solution of the original problem using the relation (10) and gives the optimal value for the original problem (1).
3 Linear-quadratic case ), and Here A (respectively B) is an n by n (respectively n by m) matrix.Observe that In this setting, the problem (1) admits a natural interpretation as a linearquadratic multi-target control problem.Its solution requires explicit computation of the coefficients of the objective function (12) which in turn requires an explicit description of orthogonal projection π Z .Such a description has been found in [FM1].We briefly describe it here.
Theorem 1 Let C be an anti-stable n by n matrix (i.e.real parts of all eigenvalues of C are positive).Consider the following system of linear differential equations: Then there exists a unique solution L(f ) of ( 15) belonging to ) is linear and bounded.Explicitly: For the proof, see [FM1] Consider the algebraic Riccati equation We assume that ( 16) has a real symmetric solution K st such that the matrix is stable (i.e.real parts of all eigenvalues of F are negative).Notice that such a solution exists if and only if the pair (A, B) is stabilizable.See e.g.[fai].
Theorem 2 Given (ψ, ϕ) ∈ H, we have x is the solution of the differential equation and ρ is a unique solution to the differential equation Remark: The required solution ρ exists and unique by Theorem 1, since the matrix −(A + BB T K st ) is anti-stable.

Sketch of the proof
Let p ∈ L n 2 [0, ∞) be absolutely continuous and such that ṗ But x(τ ), p(τ ) → 0, as τ → ∞ (see e.g.[FM1] for details) and x(0) = 0. Hence Let us now show that the decomposition ( 16), ( 18) takes place for an arbitrary (ψ, ϕ) ∈ H. Indeed, using ( 21) Hence by ( 19), ( 22) Combining all terms with x and all terms with ρ in two separate groups, we obtain Using now the fact that K st satisfies ( 16), we obtain: 20), ( 21), we obtain which is (18).Finally, it is clear that for x and u defined by ( 20), ( 21), we have ẋ = Ax + Bu and consequently (x, u) ∈ Z.This completes the proof of theorem 2. Looking at (12), we see that the evaluation of coefficients of the quadratic function requires the knowledge of expressions of the type π Z ⊥ (h) 2 , where h ∈ H. ) is the function entering the decomposition ( 17) and ( 18) and described in (22).Then Here for simplicity of notations we suppressed the dependence on t.
We can now easily compute the coefficients of the objective function ( 11).Assuming ) and noticing that by Theorem 3 where ρ(λ) is the solution of the differential equation Consequently, which allows us to easily express the objective function ( 12) in terms of integrals of ρ i and ρ c .

Concluding remarks
In this paper we have shown that multi-target linear-quadratic control problem on semi-infinite interval can be reduced to solving a simple convex optimization on the simplex.The reduction involves solving one standard algebraic Riccati equation and m + 1 linear differential equations where m is the number of targets.Notice that our results can be easily extended to discrete-time systems.