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Fig. 4. Steady-state cost as a function of Q after eliminating and .
the cost can be plotted as a function of (Fig. 4). minimizes the cost which corresponds to .
The second optimization problem (8) for the example is
where and are the optimal Lagrange multi-pliers of the respective equality constraints in the original steady-stateproblem (43). This is a quadratic cost with a positive definite Hessian
The minimizer of this problem (44) is , , and the minimizer is unique. Hence, the solutions of (43)and (44) agree, and the CSTR example satisfies strong duality.
The authors wish to thank C. Grossmann for inspiring discussions onthe necessity of proving stability of optimizing MPC schemes, and Dr.L. T. Biegler for help in developing computational methods for solvingnonlinear MPC problems.
REFERENCES J. B. Rawlings, D. Bonn, J. B. Jrgensen, A. N. Venkat, and S. B.
Jrgensen, Unreachable setpoints in model predictive control, IEEETrans. Autom Control, vol. 53, no. 9, pp. 22092215, Oct. 2008.
 S. Engell, Feedback control for optimal process operation, J. Proc.Cont., vol. 17, pp. 203219, 2007.
 M. Canale, L. Fagiano, and M. Milanese, High altitude wind energygeneration using controlled power kites, IEEE Trans. Control Syst.Tech., vol. 18, no. 2, pp. 279293, 2010.
 E. M. B. Aske, S. Strand, and S. Skogestad, Coordinator MPCfor maximizing plant throughput, Comp. Chem. Eng., vol. 32, pp.195204, 2008.
 J. V. Kadam and W. Marquardt, Integration of economical optimiza-tion and control for intentionally transient process operation, LectureNotes Control Inform. Sci., vol. 358, pp. 419434, 2007.
 J. B. Rawlings and R. Amrit, Optimizing process economic perfor-mance using model predictive control, in Nonlinear Model PredictiveControl, ser. Lecture Notes in Control and Information Sciences, L.Magni, D. M. Raimondo, and F. Allgwer, Eds. Berlin, Germany:Springer, 2009, vol. 384, pp. 119138.
 A. E. M. Huesman, O. H. Bosgra, and P. M. J. Van den Hof, Degrees offreedom analysis of economic dynamic optimal plantwide operation,in Preprints 8th IFAC Int. Symp. Dyn. Control Process Syst. (DYCOPS),2007, vol. 1, pp. 165170.
 J. V. Kadam, W. Marquardt, M. Schlegel, T. Backx, O. H. Bosgra, P.J. Brouwer, G. Dnnebier, D. van Hessem, A. Tiagounov, and S. deWolf, Towards integrated dynamic real-time optimization and controlof industrial processes, in Proc. Found. Comp. Aided Process Oper.(FOCAPO03), 2003, pp. 593596.
 D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert,Constrained model predictive control: Stability and optimality,Automatica, vol. 36, no. 6, pp. 789814, 2000.
 J. B. Rawlings and D. Q. Mayne, Model Predictive Control: Theoryand Design. Madison, WI: Nob Hill Publishing, 2009.
 E. D. Sontag, Mathematical Control Theory, 2nd ed. New York:Springer-Verlag, 1998.
 D. A. Carlson, A. B. Haurie, and A. Leizarowitz, Infinite Horizon Op-timal Control, 2nd ed. New York: Springer Verlag, 1991.
Comments on Distributed Control ofSpatially Invariant Systems
Ruth Curtain, Fellow, IEEE
The paper  was awarded the George S. Axelby Outstanding paperaward in 2004 and it has stimulated much new research into spatiallyinvariant systems, a special class of infinite-dimensional systems. Acentral theme of the paper is that the theory of spatially invariant partialdifferential systems on an infinite domain can be reduced to the study ofa family of finite-dimensional systems parameterized by , whichcan then be analyzed pointwise. The main motivating examples in were partial differential equations on an infinite domain with a spatialinvariance property which can be formulated as state linear systems on a Hilbert space as in , where generates astrongly continuous semigroup on and , , and , are also Hilbert spaces, typically
for some integer . Under mild assumptions one can take Fouriertransforms to obtain the isometrically isomorphic state linear system withthe state space , input space and output space , typically
. The new system operators are now matrices of multipli-cation operators and these are simpler to analyze. In , SectionII-B, these are seen as infinitely many finite-dimensional systems parameterized by , and theysuggest pointwise tests for various system theoretic properties of theoriginal system .
In the introduction and in Section VI.A of  the impression is giventhat the approach is applicable to many distributed control problemsincluding those involving fluid flow. The main theme of the present
Manuscript received March 01, 2010; revised July 23, 2010; acceptedNovember 14, 2010. Date of publication December 17, 2010; date of currentversion March 09, 2011. Recommended by Associate Editor K. Morris.
The author is with the Department of Mathematics, University of Groningen,Groningen 9700 AV, NL, The Netherlands (e-mail: email@example.com).
Digital Object Identifier 10.1109/TAC.2010.2099430
0018-9286/$26.00 2010 IEEE
708 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 3, MARCH 2011
paper is to point out that this approach is only applicable to a limitedsubclass of spatially invariant systems described by partial differentialequations, namely, those for which generates a -semigroup on
for some integer . Although this assumption seems innocentenough, it is actually quite restrictive. In particular, it is not satisfied fortypical second order partial differential equations.
In Section II we illustrate this by analyzing the LQR problem foran example of a scalar wave equation on an infinite domain. Althoughit is a simple spatially invariant system, in Section III we show thatthe pointwise tests in Theorems 1, 2 and 4 of  do not hold for thisexample. The reason is that it does not generate a -semigroup on
, but on another Hilbert space. Furthermore, we show that The-orems 1, 2 and 4 are direct consequences of known theory in .
In Section IV we point out that the conclusions in Section V, VI in on exponential decaying properties are also restricted to the spe-cial subclass of spatially invariant systems for which generates a-semigroup on . In particular, in Section V-A of  in thefirst paragraph of column 2 on p.1098 the authors claim that the op-timal control laws always have an inherent degree of decentralization.Moreover, that from a practical perspective, the convolution kernelscan be truncated to form local convolution kernels that have perfor-mance close to the optimal. We show that neither of these claims holdfor our wave equation example.
Section V contains some conclusions.
II. LQR CONTROL FOR THE WAVE EQUATION
Consider the following wave equation on the infinite domain :
where , . Following the approach in [1, Section IV], takingFourier transforms yields :
(Note that we prefer to take instead of .)In the approach in , Section IV, this is just a parameterized family
of finite-dimensional equations with the parameter or, writing
, equivalently with the parameter . These can be writtenin standard operator form as the system equations
on the appropriate state space. Denote
Then the appropriate state space is with theinner product
This induces the norm related to the energy of the system by
As in Example 2.2.5 in  it follows that with domain gen-erates a semigroup on . Note that it does not generate a semigroupon (c.f Exercise 2.25, ).
To solve the LQR problem for this system we need to solve thefollowing operator Riccati equation for a self-adjoint nonnegativebounded operator (see (6.56), ):
To avoid confusion we denote the adjoint with respect to by and that with respect to by . Now is self-adjoint with respectto the inner product on and this leads to the form
where , , are self-adjoint, bounded linear operators on ,i.e., they are in .
To solve this for substitute for
like components. This yields the solution
. As required, , , are all in
bounded on . Now is a bounded nonnegative operator on on if and only if the following operator is a bounded nonnegative operatoron
Its spectrum is the union of all eigenvalues of for all andit is readily verified that these are all positive.
Thus the following Lyapunov equation has a bounded positivesolution:
where . Consequently, by Theorem 5.1.3 in , weconclude that generates an exponentially stable semigroup on ,and is exponentially stabilizable.
III. APPARENT CONTRADICTIONS TO THEOREMS 1, 2, AND 4
Spatially invariant systems as considered in  are isometrically iso-morphic to state linear systems , where , , arebounded multiplication operators from to for inte-gers , . is a linear multiplication operator on , which istypically unbounded. Furthermore, it is asssumed that , ,, are continuous for . (cf. Assumption 1 in ).Our example in Section II satisfies all these assumptions.
Let us recall the Lyapunov Theorem 1 in : If is the generatorof a semigroup, then it is exponentially stable if and only if for each the Lyapunov equation
(3)has a solution with components that are bounded for all .