Multitarget Linear-Quadratic Control Problem:

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Volume 2012, Article ID 535610,10pages doi:10.1155/2012/535610

Research Article

Multitarget Linear-Quadratic Control Problem:

Semi-Infinite Interval

L. Faybusovich

¹

and T. Mouktonglang

^{2, 3}

1Mathematics Department, University of Notre Dame, Notre Dame IN 46556, USA

2Mathematics Department, Faculty of Science, Chiang Mai University, Chiang Mai 50220, Thailand

3Centre of Excellence in Mathematics, CHE, Sri Ayutthaya Road, Bangkok 10400, Thailand

Correspondence should be addressed to T. Mouktonglang,[email protected] Received 12 September 2011; Accepted 13 October 2011

Academic Editor: Ion Zaballa

Copyrightq2012 L. Faybusovich and T. Mouktonglang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

LetH,,be a Hilbert space,Zbe its closed vector subspace,h1, . . . , hm, andcbe vectors in H. Consider the following optimization problem:

1≤i≤mmaxh−hi −→min, h∈cZ. 1.1

Here · is the norm in H induced by the scalar product,. In1, we analyzed 1.1 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 multitarget linear-quadratic control problem on a finite time interval. In this paper, we consider a reduction1.1to even simpler optimization problem of minimization of convex quadratic function on them−1dimensional simplex. We then apply this result to the analysis of a multitarget 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.

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2. Reduction to a Simple Quadratic Programming Problem

Letfih h−hi², i 1,2, . . . , m. It is obvious that1.1is equivalent to the following optimization problem:

z−→min,

fih≤z, i1,2, . . . , m , h∈cZ.

2.1

Consider the Lagrange function

Lλ1, . . . , λm, h, z z^m

i1

λi

fih−z

z

1−^m

i1

λi

^m

i1

λifih.

2.2

Notice that despite the fact that our original problem is infinite dimensional, the usual KKT theorem holds true see e.g., 2, page 72. It is also clear that Slater conditions are satisfied. Hence, optimality condition for2.1takes the form

λi≥0, λi

fih−z

0, i0,1,2, . . . , m,

∂L

∂z 0, ^m

i1λi∇fih∈Z^⊥, 2.3

where ∇fih 2h−hi, i 1,2, . . . , m, Z^⊥ is the orthogonal complement of Z in H.

Conditions2.3lead to

m i0

λi1, λi≥0, i1,2, . . . , m,

πZh ^m

i1λiπZhi.

2.4

HereπZ:H → Zis the orthogonal projection. Let us form the Lagrange dual of2.1.

Consider

ϕλ1, λ2, . . . , λm min{Lλ1, . . . , λm, h, z:h∈cZ, z∈Z}. 2.5

Using2.4, we obtain that

ϕλ1, λ2, . . . , λm

m i1

λifihλ1, . . . , λm, 2.6

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where

hλ1, . . . , λm πZ^⊥c ^m

i1

λiπZhi. 2.7

Notice that for anyh∈cZ, πZ^⊥h πZ^⊥c. HereπZ^⊥:H → Z^⊥is the orthogonal projection ofHonto orthogonal complementZ^⊥ofZ. To further simplify2.6, introduce the notation

hλ ^m

i1

λihi. 2.8

Then

fjhλ1, . . . , λm πZ

hλ−hj

πZ^⊥

c−hj² πZ

hλ−πZ

hj²πZ^⊥

c−hj² πZhλ²πZ

hj²−2 πZhλ, πZ

hj

πZ^⊥

c−hj².

2.9

Hence, according to2.6, we have the following:

ϕλ1, . . . , λm πZhλ²^m

j1

λjπZ

hj²

−2πZhλ, πZhλ^m

j1

λjπZ^⊥

c−hj².

2.10

We, hence, arrive at the following expression ofϕ:

ϕλ1, . . . , λm − πZ

_m

i1

λihi

2

^m

j1

λj

πZ

hj²πZ^⊥

c−hj²

. 2.11

We can simplify2.11somewhat. Notice that πZ^⊥

c−hj²πZ^⊥c²πZ^⊥

hj²−2 πZ^⊥c, πZ^⊥

hj

. 2.12

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Consequently,

ϕλ1, . . . , λm −πZhλ²^m

j1

λjhj²

−2πZ^⊥c, πZ^⊥hλπZ^⊥c² −hλ²πZ^⊥hλ−c²^m

j1

λjhj².

2.13

Here,

hλ ^m

i1

λihi. 2.14

Hence, the Lagrange dual to2.1takes the following form:

ϕλ1, . . . , λm−→max, m

i1λi1, λi ≥0, i1,2, . . . , m. 2.15 Ifλ^∗₁, . . . , λ^∗_mis an optimal solution to2.15, we can recover the optimal solution of the original problem using the relation2.7, andϕλ^∗₁, . . . , λ^∗_mgives the optimal value for the original problem1.1.

3. Linear-Quadratic Case

Denoted byLⁿ₂0,∞, the vector space of square integrable functionsf : 0,∞ → Rⁿ. Let HLⁿ₂0,∞×L^m₂0,∞, and

Z

α, β

∈H:αis absolutely continuous on0,∞, α˙ AαBβ, α0 0

. 3.1

HereArespectivelyBis annbynrespectivelynbymmatrix. Observe that

α1, β1

, α2, β2

_∞

0

α1t^Tα2t β1t^Tβ2t dt, αi, βi

∈H, i1,2.

3.2

In this setting, the problem1.1admits a natural interpretation as a linear-quadratic multitarget control problem. An interesting solution for this problem form 2 is described in3. In our approach, we need an explicit computation of the coeﬃcients of the objective function 2.13 which in turn requires an explicit description of orthogonal projectionπZ. Such a description has been found in4. We briefly describe it here.

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Theorem 3.1. LetCbe an antistablenbynmatrix (i.e., real parts of all eigenvalues ofCare positive).

Consider the following system of linear diﬀerential equations:

x˙ Cxf, 3.3

where f ∈ Lⁿ₂0,∞. Then there exists a unique solution Lf of 3.3 belonging to Lⁿ₂0,∞.

Moreover, the mapL:Lⁿ₂0,∞ → Lⁿ₂0,∞is linear and bounded. Explicitly,

L f

t − _∞

0

e^−Cτftτdτ. 3.4

For the proof, see4.

Consider the algebraic Riccati equation

KBB^TKA^TKKA−I0. 3.5

We assume that3.5has a real symmetric solutionKstsuch that the matrix

FABB^TKst 3.6

is stablei.e., real parts of all eigenvalues ofFare negative. Notice that such a solution exists if and only if the pairA, Bis stabilizable. See, for example,5.

Theorem 3.2. We have the following:

Z^⊥

p˙A^Tp, B^Tp

; p∈Lⁿ₂0,∞, pis absolutely continuous, p˙∈Lⁿ₂0,∞

. 3.7

Given thatψ, ϕ∈H, we have

ψx−

p˙A^Tp

, 3.8

ϕu−B^Tp, 3.9

wherexis the solution of the diﬀerential equation

x˙

ABB^TKst

xBB^TρBϕ, x0 0, 3.10

uB^TKstxB^Tρϕ, 3.11

pKstxρ, 3.12

andρis a unique solution to the diﬀerential equation

ρ˙−

ABB^TKst

T

ρ−KstBϕ−ψ 3.13

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belonging toLⁿ₂0,∞.

In particular,x, u∈Z, −p˙A^Tp, B^Tp∈Z^⊥, and consequentlyZis a closed subspace in Hwith

πZ

ψ, ϕ

x, u, πZ^⊥

ψ, ϕ −

p˙A^Tp, B^Tp

. 3.14

Remark 3.3. The required solutionρexists and unique byTheorem 3.1, since the matrix−A BB^TKstis antistable.

Sketch of the Proof

Letp∈Lⁿ₂0,∞be absolutely continuous and such that ˙p∈Lⁿ₂0,∞. Suppose thatx, u∈Z.

Then

x, u,

p˙A^Tp, B^Tp

_∞

0

x^Tp˙x^TA^TpuB^Tp dt

_∞

0

x^Tp˙ AxBu^Tp dt

_∞

0

x^Tp˙x˙^Tp dt

_∞

0

d dt

x^Tp

dt lim

τ→ ∞x^Tτpτ−x0^Tp0.

3.15

Butxτ, pτ → 0, asτ → ∞see e.g.,4for detailsandx0 0. Hence, x, u,

p˙A^Tp, B^Tp

0. 3.16

Let us now show that the decomposition3.5and3.9takes place for an arbitrary ψ, ϕ∈H. Indeed, using3.12,

p˙ Kstx˙ρ.˙ 3.17

Hence by3.10and3.13,

p˙Kst

ABB^TKst

xKstBB^TρKstBϕ−

ABB^TKst

T

ρ−KstBϕ−ψ. 3.18

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Combining all terms withxand all terms withρin two separate groups, we obtain that

p˙A^Tpp˙A^TKstxA^Tρ

KstAKstBB^TKstA^TKst

x

KstBB^T−A^T−KstBB^TA^T ρ−ψ.

3.19

Using now the fact thatKstsatisfies3.5, we obtain that

p˙A^Tpx−ψ 3.20

which is3.8. Using3.11and3.12, we obtain that

u−B^TpB^TKstxB^Tρϕ−B^TKstx−B^Tρ

ϕ, 3.21

which is3.9. Finally, it is clear that forxandudefined by3.11and3.12, we have

x˙ AxBu 3.22

and consequentlyx, u∈Z. This completes the proof ofTheorem 3.2.

Looking at2.13, we see that the evaluation of coeﬃcients of the quadratic function requires the knowledge of expressions of the typeπZ^⊥h², whereh∈H.

Theorem 3.4. Leth ψ, ϕ ∈ H, andρ ∈ Lⁿ₂0,∞is the function entering the decomposition 3.8and3.9and described in3.13. Then

πZh²B^Tρϕ², 3.23 πZ^⊥h²h²−B^Tρϕ². 3.24

Proof. Lety, ν∈Z. Let, further,

Δ y, ν

ν−B^TKsty−B^Tρ−ϕT

ν−B^TKsty−B^Tρ−ϕ

. 3.25

Here for simplicity of notations, we suppressed the dependence ont. Then Δ

y, ν

Δ1 Δ2 Δ3, 3.26

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where Δ1

ν−ϕT ν−ϕ

, Δ2

KstyρT

BB^T

Kstyρ

, andΔ3−2

ν−ϕT

B^TKstyρ . 3.27

Sincey, ν∈Z, we have

y˙ AyBν, y0 0. 3.28

Hence,

Δ2 y^T

KstBB^TKst

yρ^TBB^Tρ2ρ^TBB^TKsty,

Δ3−2

Bν−Bϕ_T

Kstyρ −2

y˙−Ay−Bϕ_T

Kstyρ −2 ˙yKstyy^T

A^TKstKstA y2

BϕT

Ksty

−2 ˙y^Tρ2 AyT

ρ2 BϕT

ρ.

3.29

Notice that ˙y^Tρy^Tρ˙ d/dty^Tρ. Hence, Δ

y, ν

ν−ϕT ν−ϕ

y^T

KstBB^TKstA^TKstKstA y

2y^T

ρ˙KstBϕKstBB^TρA^Tρ

B^TρT B^Tρ

2ϕ^T B^Tρ

− d dt

y^Tρ

− d dt

y^TKsty .

3.30

Using the fact thatKstis a solution to3.5and3.13, we obtain that Δ

y, ν

ν−ϕT ν−ϕ

y^Ty−2y^Tψ

B^Tρϕ_T

B^Tρϕ

−ϕ^Tϕ− d dt

y^Tρ

− d dt

y^TKsty

ν−ϕ_T ν−ϕ

y−ψ_T y−ψ

B^TρϕT

B^Tρϕ

−ϕ^Tϕ−ψ^Tψ− d dt

y^Tρ

− d dt

y^TKsty .

3.31

Integrating3.31from 0 to∞and using the fact thaty0 0, yt, ρt → 0 ast → ∞, we obtain that

_∞

0

Δ y, ν

dty−ψ, ν−ϕ²−ψ, ϕ²B^Tρϕ². 3.32

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Notice thatΔy, ν≥0 andΔy, ν 0 providedy, ν πZψ, ϕ. See3.11. Consequently, 3.32implies that

ψ, ϕ²B^Tρϕ²πZ^⊥

ψ, ϕ

². 3.33

Hence,

πZ

ψ, ϕ²B^Tρϕ². 3.34

This completes the proof ofTheorem 3.4.

We can now easily compute the coeﬃcients of the objective function2.11. Assuming thathi ψi, ϕi∈Lⁿ₂0,∞×L^m₂0,∞, i1,2, . . . , m, c α, β∈Lⁿ₂0,∞×L^m₂0,∞and noticing that byTheorem 3.4

πZhλ−c² _∞

0

B^Tρλ ϕλ_T

B^Tρλ ϕλ

dt, 3.35

whereρλis the solution of the diﬀerential equation d

dtρλ −

ABB^TKst

_T

ρλ−KstB

ϕλ−ψλ

, 3.36

belonging toLⁿ₂0,∞and

ϕλ ^m

i1

λi

ϕi−β

, ψλ ^m

i1

λi

ψi−α

. 3.37

Consequently,

ρλ ^m

i1

λi

ρi−ρc

, 3.38

whereρiandρcareLⁿ₂0,∞solutions of diﬀerential equations ρ˙i−

ABB^TKst

ρi−KstBϕi−ψi, i1,2, . . . , m, ρ˙c−

ABB^TKst

ρc−KstBβ−α,

3.39

respectively.

Hence,

πZhλ−c² _∞

0

Γλ^TΓλdt, 3.40

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where

Γλ ^m

i1

λi

B^T

ρi−ρc

ϕi−β

, 3.41

which allows us to easily express the objective function2.13in terms of integrals ofρiand ρc.

4. Concluding Remarks

In this paper, we have shown that multitarget 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 m1 linear diﬀerential equations, wheremis the number of targets. Notice that our results can be easily extended to discrete-time systems.

Acknowledgments

The research of L. Faybusovich was partially supported by NSF Grant DMS07-12809.

The research of T. Mouktonglang was partially supported by the Commission on Higher Education and Thailand Research Fund under Grant MRG5080192.

References

1 L. Faybusovich and T. Mouktonglang, “Multi-target linear-quadratic control problem and second- order cone programming,” Systems & Control Letters, vol. 52, no. 1, pp. 17–23, 2004.

2 G. G. Magaril-Il’yaev and V. M. Tikhomirov, Convex Analysis: Theory and Applications, vol. 222 of Trans- lations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 2003.

3 A. S. Matveev and V. A. Yakubovich, Optimal Control Systems, Petersburg University, St. Petersburg, Russia, 2003.

4 L. Faybusovich and T. Mouktonglang, “Linear-quadratic control problem with a linear term on semi- infinite interval: theory and applications,” Tech. Rep., University of Notre Dame, December 2003.

5 L. E. Fa˘ıbusovich, “Algebraic Riccati equation and symplectic algebra,” International Journal of Control, vol. 43, no. 3, pp. 781–792, 1986.

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