## Abstract

We formulate stochastic robust optimal control problems, motivated by applications arising in interconnected economic systems, or spatially extended economies. We study in detail linear quadratic problems and nonlinear problems. We derive optimal robust controls and identify conditions under which concerns about model misspecification at specific site(s) could cause regulation to break down, to be very costly, or to induce pattern formation and spatial clustering. We call sites associated with these phenomena hot spots. We also provide an application of our methods by studying optimal robust control and the potential break down of regulation, due to hot spots, in a model where utility for in situ consumption is distance dependent.

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## Notes

- 1.
- 2.
- 3.
Although we choose to interpret the characteristics associated with the distributed parameter aspect as physical space, the notion of “space” does not have to be physical. It can be used to model characteristics that are associated with economic, sociological, cultural, or other factors. Since the notion of “space” may be broadly interpreted, this suggests that our methods can be used for the analysis of a wide range of problems.

- 4.
This choice is for simplicity of presentation. Most of the arguments and results presented here can be extended to infinite dimensional systems, admittedly with considerable technical effort employing techniques beyond the scope of the present paper, or when explicitly stated so with a weighted version of this space.

- 5.
The generalization to vector valued state and control variables \(x^{n}\in {\mathbb {R}}^{d_{1}}\) and \(u_{n}\in {\mathbb {R}}^{d_{2}}\) on each site \(n\in {\mathbb {Z}}\) requires the use of the sequence spaces \(\ell ^{2}({\mathbb {R}}^{d_{i}})\), \(i=1,2\) rather than \(\ell ^{2}:=\ell ^{2}({\mathbb {R}})\)) and is straightforward. Furthermore, the generalization for infinite dimensional lattices is feasible but becomes technical from the mathematical points of view and is beyond the scope of the present paper.

- 6.
There is uncertainty concerning the economy which is represented in terms of the vector valued stochastic process \(w\). These common factors affect the state of the economy \(x\) at the different sites. Each factor has a different effect on the state of the economy on each particular site; this will be modeled by a suitable correlation matrix. It is not of course necessary that the number of factors is the same as the number of sites in the system; however, without loss of generality we will make this assumption and assume that there is one factor or source of uncertainty related to each site. This assumption can be easily relaxed.

- 7.
In the limiting case where \(v=\{v_{n}\}\) is a constant vector this leads to \({\mathbb {Q}}( w(t) \in A)=\int _{A} (2 \pi t)^{-\frac{N}{2}} \exp \left( -\frac{||x-v t||^2}{2 t} \right) dx\), i.e., \(w\) is distributed according to the normal law \({\mathcal {N}}(v \, t,{\mathsf {I}} \,t)\).

- 8.
Girsanov’s theorem is a very powerful result in stochastic analysis describing how the law of a Wiener process is transformed when viewed under an alternative probability measure, and is essentially a change of drift argument, finding important applications in mathematical finance, stochastic control theory, mathematical economics, etc. The validity of the theorem requires some technical assumptions (a) regarding the filtration \(\{\mathcal{F}_{t}\}_{t \in [0,T]}\) often quoted as the usual conditions, i.e., right continuity and the property that \(\mathcal{F}_{0}\) contains all the null sets of \({\mathbb {P}}\) and (b) regarding the information drift process \(v\), which essentially is required to make sure that the stochastic exponential \(\mathcal{E}_{t}(v)\) (defined through (2) by setting \(T=t\)) is a martingale, rather than just a local martingale. The Novikov condition is a general condition that guarantees (b). For an excellent, clear and detailed exposition of Girsanov’s theorem see [27].

- 9.
A matrix \({\mathsf {A}}\) is translation invariant if \({\mathsf {A}}{\mathsf {T}}_{k} = {\mathsf {T}}_{k} {\mathsf {A}}\) where \({\mathsf {T}}\) is the translation by \(k\) on \(\ell ^{2}({\mathbb {Z}}_{N}) \simeq {\mathbb {R}}^{N}\).

- 10.
that correspond to the first column of the matrices \({\mathsf {A}}\), \({\mathsf {B}}\), and \({\mathsf {C}}\)

- 11.
One might consider as a breakdown of solutions the absence of positive real roots which would correspond to a convex value function for the game (see Remark 2). However, the absence of real roots, whatsoever, corresponds to an even worse situation which implies that the HJBI equation is ill posed and admits no solutions at all, at least of the quadratic type.

- 12.
The Haar measure is a generalization of the Lebesgue measure, which is invariant under the symmetry group

- 13.
The general form \({\mathsf {F}} (x)=(f_{1}(x),f_{2}(x),\ldots ) \) where at each site the nonlinear effects depend on the state of the system at all lattice sites is easily handled within the framework presented here.

- 14.
In this particular case \({\mathbb {H} }=\ell ^{2}:=\ell ^{2}({\mathbb {Z}}_{N})\simeq {\mathbb {R}}^{N}\) and by the finite dimensionality of the Hilbert space involved \({\mathbb {H}}={\mathbb {H} }^{*}\).

- 15.
The full nonlinear model can be considered in the context of Sect. 6 with similar qualitative result. The linear approach provides, however, better tractability.

- 16.
See, for example, [5] for calibrating the parameter \(\theta \) using scientific information related to climate change.

## References

- 1.
Akamatsu T, Takayama Y, Ikeda K (2012) Spatial discounting, fourier, and racetrack economy: a recipe for the analysis of spatial agglomeration models. J Econ Dyn Control 36(11):1729–1759

- 2.
Albeverio S, Makarov K, Motovilov A (2003) Graph subspaces and the spectral shift function. Canad. J. Math 55(3):449–503

- 3.
Anderson E, Hansen L, Sargent T (2003) A quartet of semigroups for model specification, robustness, prices of risk, and model detection. J Eur Econ Assoc 1(1):68–123

- 4.
Armaou A, Christofides PD (2001) Robust control of parabolic pde systems with time-dependent spatial domains. Automatica 37(1):61–69

- 5.
Athanassoglou, S. and A. Xepapadeas (2011). Pollution control: when, and how, to be precautious. Fondazione eni enrico mattei working papers, 569.

- 6.
Aubin J, Ekeland I (1984) Applied nonlinear analysis. Wiley, New York

- 7.
Bamieh B, Paganini F, Dahleh M (2002) Distributed control of spatially invariant systems. IEEE Trans Automat Contr 47(7):1091–1107

- 8.
Başar T, Bernhard P (2008) H\(^{\infty }\)-optimal control and related minimax design problems: a dynamic game approach. Birkhauser, Boston

- 9.
Bensoussan A, Da Prato G, Delfour M, Mitter S (1992) Representation and control of infinite dimensional systems: Vol 2 (systems & control: foundations & applications). Birkhäuser Boston Inc., Boston, MA

- 10.
Boucekkine R, Camacho C, Zou B (2009) Bridging the gap between growth theory and the new economic geography: the spatial ramsey model. Macroecon Dyn 13:20–45

- 11.
Brito, P. (2004). The dynamics of growth and distribution in a spatially heterogeneous world. Instituto Superior de Economia e Gestão - DE Working papers n\(^\circ \) 14-2004/DE/UECE

- 12.
Brock W, Xepapadeas A (2008) Diffusion-induced instability and pattern formation in infinite horizon recursive optimal control. J Econ Dyn Control 32(9):2745–2787

- 13.
Brock W, Xepapadeas A (2010) Pattern formation, spatial externalities and regulation in coupled economic-ecological systems. J Environ Econ Manage 59:149–164

- 14.
Cerrai S (2001) Second order PDE’s in finite and infinite dimension: a probabilistic approach. Springer Verlag, Berlin

- 15.
Curtain, R., O. Iftime, and H. Zwart (2008). System theoretic properties of platoon-type systems. In Decision and control, 2008. CDC 2008. 47th IEEE Conference on decision and control , p 1442–1447. IEEE.

- 16.
Da Prato G (2002) Linear quadratic control theory for infinite dimensional systems. . Mathematical control theory. ICTP Lecture notes, vol 8. p 59–105

- 17.
Da Prato G, Zabczyk J (2002) Second order partial differential equations in Hilbert spaces. London mathematical society lecture notes. Cambridge Univ Pr., Cambridge

- 18.
Desmet K, Rossi-Hansberg E (2010) On spatial dynamics. J Reg Sci 50(1):43–63

- 19.
Gilboa I, Schmeidler D (1989) Maxmin expected utility with non-unique prior. J Math Econ 18:141–153

- 20.
Haldane A (2009) Rethinking the financial network. Speech delivered at the Financial Student Association, Amsterdam

- 21.
Hansen L, Sargent T (2001) Robust control and model uncertainty. Am Econ Rev 91:60–66

- 22.
Hansen L, Sargent T (2008) Robustness in economic dynamics. Princeton university Press, Princeton

- 23.
Hansen L, Sargent T, Turmuhambetova G, Williams N (2006) Robust control and model misspecification. J Econ Theory 128:45–90

- 24.
Isaacs R (1965) Differential games: A mathematical theory with applications to welfare and pursuit, control and optimization. Wiley, New York

- 25.
JET (2006) Symposium on model uncertainty and robustness. J Econ Theory 128:1–163

- 26.
Johnson CR (1989) A gersgorin-type lower bound for the smallest singular value. Linear Algebra Appl 112:1–7

- 27.
Karatzas I, Shreve S (1991) Brownian motion and stochastic calculus. Springer, New York

- 28.
Karatzas I, Shreve SE (1998) Methods of mathematical finance. Springer, New York

- 29.
Krugman P (1996) The self-organizing economy. Blackwell, Oxford

- 30.
Leizarowitz A (2008) Turnpike properties of a class of aquifer control problems. Automatica 44(6):1460–1470

- 31.
Ljungqvist L, Sargent T (2004) Recursive macroeconomic theory. The MIT Press, Cambridge

- 32.
Magill M (1977) A local analysis of n-sector capital accumulation under uncertainty. J Econ Theory 15(1):211–219

- 33.
McMillan C, Triggiani R (1994) Min-max game theory and algebraic riccati equations for boundary control problems with continuous input-solution map. part ii: the general case. Appl Math Optim 29(1):1–65

- 34.
Murray J (2003) Mathematical biology, Vol. I and II, 3rd edn. Springer, New York

- 35.
Perrings C, Hannon B (2001) An introduction to spatial discounting. J Reg Sci 41(1):23–38

- 36.
Rudin W (1990) Fourier analysis on groups. Wiley-Interscience, New York

- 37.
Salmon M (2002) Special issue on robust and risk sensitive decision theory. Macroecon Dyn 6(1):19–39

- 38.
Sanchirico J, Wilen J (1999) Bioeconomics of spatial exploitation in a patchy environment. J Environ Econ Manage 37(2):129–150

- 39.
Smith M, Sanchirico J, Wilen J (2009) The economics of spatial-dynamic processes: applications to renewable resources. J Environ Econ Manage 57:104–121

- 40.
Smith T (1975) An axiomatic theory of spatial discounting behavior. Pap Reg Sci 35(1):31–44

- 41.
Smith T (1976) Spatial discounting and the gravity hypothesis. Reg Sci Urban Econ 6(4):331–356

- 42.
Turing A (1952) The chemical basis of morphogenesis. Philos Trans R Soc Lond 237:37–72

- 43.
Whittle P (1996) Optimal control: basics and beyond. Wiley, New York

- 44.
Wilen J (2007) Economics of spatial-dynamic processes. Am J Agric Econ 89(5):1134–1144

- 45.
Wong M (2011) Discrete fourier analysis. Birkhauser, Boston

- 46.
Wu J, Plantinga A (2003) The influence of public open space on urban spatial structure. J Environ Econ Manage 46(2):288–309

- 47.
Zhu Y, Pagilla PR (2005) A note on the necessary conditions for the algebraic riccati equation. IMA J Math Control Inf 22(2):181–186

## Acknowledgments

This research has been co-financed by the European Union (European Social Fund - ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: “Thalis - Athens University of Economics and Business - Optimal Management of Dynamical Systems of the Economy and the Environment: The Use of Complex Adaptive Systems”. W. Brock is grateful for financial and scientific support received from The Center for Robust Decision Making on Climate and Energy Policy (RDCEP) which is funded by a grant from the National Science Foundation (NSF) through the Decision Making Under Uncertainty (DMUU) program. He is also grateful to the Vilas Trust for financial support. None of the above is responsible for any errors, opinions, or shortcomings in this article. The authors are indebted to one of the anonymous referees whose thorough and critical reading of the manuscript and insightful comments highly contributed to the improvement of the current version of this work.

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### Cite this article

Brock, W.A., Xepapadeas, A. & Yannacopoulos, A.N. Robust Control and Hot Spots in Spatiotemporal Economic Systems.
*Dyn Games Appl* **4, **257–289 (2014). https://doi.org/10.1007/s13235-014-0109-z

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### Keywords

- Stochastic control
- Robust control
- Riccati equation
- Hot spot formation
- Pattern formation
- Hamilton-Jacobi Bellman equation
- Distance-dependent utility

### JEL Classification

- C6
- R12
- Q26