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This survey highlights some recent advances on modeling the economic state of some nations of the world which are linked up in solidarity and competition or which are isolated. The problem of controllability, convergence, and permanence are explored. Some practical recent policy implications are deduced. Earth revolves around the sun in approximately circular orbit with radius r = a , completing a revolution in the time T = 2 π ( a 3 / GM ) 1 / 2 which is one earth year; here M is the mass of the sun and G is the universal gravitational constant. The gravitational force of the sun on the earth is given by GM m / r 2 where m is the mass of the earth. Therefore, if the earth “stood still” losing its orbital velocity, it will fall straight on a line into the sun in accordance with Newton's second law. M d 2 r d t 2 = - GM M r 2 . If this calamity occurred what fraction of the normal year T will it take for the earth to splash into the sun?” (Nagle, Saff, Snider) “Why did the earth stand still?” (Ndu Udo) “The symbol of Love and the earth's lover is the moon. With joy it dances around the earth in a circular orbit. When the inhabitants of the earth prefer hate and revenge to love, the moon stops to love, and stands still, losing its orbital velocity. It falls straight on a line into the earth in accordance with Newton's second law. The crash makes the earth stand still precipitating the Armageddon.” (Wisdom Goodness)

In this paper, we survey the mathematical theory of competitive an co-operative systems that monitor national interactions and economic systems. The dynamics are described by ordinary differential equations, functional differential equations and coupled partial differential equations. Conditions are given which guarantee finite time extinction, finite time unbounded growth, and persistence. These conditions are already available in the literature (Proceedings of the First World Congress of Nonlinear Analysis, 1992 Vol I–IV 359–368 Gruyler, Berlin; Appl. Anal. 57 (1995) 3–4, pp. 309–323; Math. Biosci. 118 (1993) 197; In: T.G. Hallam, S.A. Levin (Eds.), Mathematical Ecology, Biomathematics, Vol. 17, Biomathematics, Springer, Berlin, Heidelberg, 1986; Convex Structures and Economic Theory, Academic Press, New York, 1968; SIAM J. Math. Anal. 18 (1987) 642; Introduction to Differential Equation, Prentice-Hall, Englewood Cliffs, NJ, 1987; SIAM J. Appl. Math. 36 (1979) 421; Proc. Nat. Acad. Sci. USA 68 (1971) 980; SIAM J. Math. Anal. 24 (1993) 1331; The Passionate God, Paulist Press, New York, 1981; Nonlinear Parabolic and Elliptic Equation, Plenum Press, New York, 1992; Economic Theory and Social Justice, MacMillan, London, UK, 1999; Extinction in finite time of solutions to nonlinear absorption–diffusion equations, personal communication). The theory is applied to national wealth that is carefully defined. It is shown that the wealth of co-operating nations can grow unbounded and competing ones become extinct. Using the principle of “trickle down of wealth” and incorporating a strategy of internally generated wealth due to improved health and education, we derive the dynamics of wealth.

VOLTERRA INTEGODIFFERENTIAL NEUTRAL DYNAMICS FOR THE GROWTH OF WEALTH OF NATIONS : A CONTROLLABILITY THEORY

In this section we derive a hereditary model of gross domestic product, y, and employment L by invoking the “demand and supply principle”, and the “rational expectations hypothesis” which assumes that the expected values of economic variables are functions of the current and past values. Indeed let the aggregate demand function z i be given by $$z_i = I_i + C_i + X_i + G_i,\quad i = 1, 2$$ (7.1)

In this chapter models of Gross Domestic Product and Employment of two interacting nations are derived. This system is a hereditary differential game of pursuit. It utilizes the principles of supply and demand and of rational expectations. A theory of interaction via net export is postulated. The net export function may include trade, debt repayment, etc. International employments can also be included. The UN and IMF data for two nations are used to validate the theory and test for the degree of cooperation and competition. MATLAB andMaple programs are exploited. The consequences of cooperation and competition are studied, and controllability questions answered. Once the system is proved controllable it can be made sufficiently cooperating, and sustained growth of GDP and employment can be guaranteed.

The economic state is x = [y, R, L, k, p, E). We now recall our earlier equations for interest rate [2, p. 29] – Stability and Time Optimal Control of Hereditary Systems with Applications to Economic Dynamics of the USA. Let M denote money supply (e.g., M1) $$\overline{L} = {\rm money\,demand} \approx {\rm Md}$$

Aggregate demand zi is given by $$z_i = I_i + C_i + X_i + G_i,\quad i = 1,...,191,$$ (2.1) is the contribution to the country of yi interacting with countries of yj (j ≠ i) due to cooperation and competition.

Very recently the President of the USA, set up a bipartisan commission to generate strategies to boost the positive value of the Balance of Payment of the USA. A mathematical solution can be deduced from the following research monographs of the author, E. N. Chukwu. (1) E. N. Chukwu, Stability and Time Optimal Control of Hereditary Systems with Application to the Economic Dynamics of the US, 2nd edition, World Scientific, 2001, ISBN 981-02-4674-9. Singapore, New Jersey, London. (2) E. N. Chukwu, Differential Models and Neutral Systems for Controlling the Wealth of Nations, World Scientific, 2001, ISBN 9810243812, Singapore, New Jersey, London. (3) Optimal Control of the Growth of Wealth of Nations, Taylor and Francis, 2003, ISBN 0-415-26966-0, London, New York. (4) E. N. Chukwu, A Mathematical Treatment of Economic Cooperation and Competition Among Nations with Nigeria, USA, UK, China and Middle East Examples, Mathematics in Science and Engineering, v 203, Elsevier, 2005, ISBN-13978-0-444-51859-0, Amsterdam, The Netherlands. (5) E. N. Chukwu, The Omega Problem of all members of the United Nations, Atlantis Studies in Mathematics for Engineering and Science, v 7, World Scientific, pub. Date scheduled Summer 2010. (6) E. N. Chukwu, Stability and Time Optimal Control f Hereditary Systems, Academic Press, Boston, 1992.

Employment is currently decaying in the United States of America. Policies of private firms and government to arrest the situation are now hotly debated. This paper is a readable reflection and extraction from Chukwu’s books, Stability and Time-Optimal Control of Hereditary Systems with Application to the Economic Dynamics of the USA,” 2nd Edition, World Scientific 2001 [1], and “The Omega Problem of all Members of the United Nations”, Atlantis Studies in Mathematics for Engineering and Science, Volume 6. Series editor, C. K. Chui, Stanford University, U.S.A. and World Scientific [2]. Reasonable and verifiable economic policies are deduced from Chukwu.

The control of economic system's growth with delay is formulated to illustrate Euclidean (E^n) and function space (W"2^(^1^)) controllability of a general nonlinear delay equation. Sufficient conditions are given, in terms of the control matrix, for controllability of the general nonlinear system. These conditions are shown to be necessary for W"2^(^1^) controllability. The theory is then applied to a simple dynamics of gross national product and interest rate. The concept of function space controllability is very appropriate: once the target is hit, it remains on a certain growing function for some time.

In this paper, we show that the growth of isolated capital stock or national income can be described as a nonlinear hereditary system. When n such systems are linked up to others in the region by a solidarity function which may be taken to be the effect of government and other factors on the subsystem of n firms, we use investment, export, money holding, and consumption strategies as the firms' vector instruments to control the growth of capital stock. Government vector instruments include nominal value of money, taxation, and investment (for example, into innovation). The control problem is viewed as a game of pursuit between government and the firms. Proven theorems provide universal laws for the growth of capital stock from an initial endowment to a target. Though the control strategy constructed to reach the target may not be usable, except when linear or if nonlinear, under some exceptional circumstances because the fundamental matrix is too difficult to construct, the consequences nevertheless provide a solution to this century's burning problem of how much government intervention (compared with private firms) should be to ensure economic growth. For this alone, i.e., (6.8), (6.9), (6.10), the research here seems important. A very simple model of the Austrian economy is presented.

IfVis a Lyapunov function of an equationdu/dt=u′=Zuin a Banach space then asymptotic stability of an equilibrium point may be easily proved if it is known thatsup(V′)<0on sufficiently small spheres centered at the equilibrium point. In this paper weak asymptotic stability is proved for a bounded infinitesimal generatorZunder a weaker assumptionV′≤0(which alone implies ordinary stability only) if some observability condition, involvingZand the Frechet derivative ofV′, is satisfied. The proof is based on an extension of LaSalle's invariance principle, which yields convergence in a weak topology and uses a strongly continuous Lyapunov function. The theory is illustrated with an example of an integro-differential equation of interest in the theory of chemical processes. In this case strong asymptotic stability is deduced from the weak one and explicit sufficient conditions for stability are given. In the case of a normal infinitesimal generatorZin a Hilbert space, strong asymptotic stability is proved under the following assumptionsZ*+Zis weakly negative definite andKer Z={0}. The proof is based on spectral theory.

In this paper, we carefully derive the dynamics of the growth of national income as a functional differential equation. We consider autonomous consumption, export, investment and real money demand as “private control variable” or pursuer strategy. We define as “solidarity function” the quarry strategy or government control instrument, government expenditure, taxation, and nominal value of money supply, and consider our dynamics as a differential game of pursuit. Conditions for oscillation (boom and depression) will be given. Optimal strategy to attain a desired growth of income or employment rate will be deduced from the theory of optimal control of a hereditary system.

The function \(p\in L_{\infty}([\sigma,t],P), g \in L_{\infty}([\sigma,t],Q)\) are called the private firms’ control strategy and the governments’ (solidarity) strategy functions or controls respectively. The strategies p and g are used to steer initial value ψ ≡ x 0 of the nations’ gross domestic to a target x1 in time t1.

In the previous chapter we studied economic growth by modeling through MATLAB regression the growth of gross domestic product, interest rate, employment, value of capital stock, prices, and cumulative balance of payment. It is now a common observation that there is a 30-year gain in the last century of longevity which is related to good health and generation of wealth. Dr. Robert Butler of the International Longevity Center who has studied this for over fifty years affirms this 30-year gain. Furthermore those nations that have a five-year advance in longevity compared to other countries actually have a greater GDP.

In Chapter 9, Program Result for Some Nations: Austria and France, we display samples of what can be generated from the data, the programs and their execution from the contents of CD-Rom of the 141 nations. The identified government strategies and the equation of the representative firms strategies yield testable policies for the economic growth of each nation.

Just as in Botwana2.m policies can be deduced from the equation for y. It was executed by E. Chukwu. The raw data was taken from International Yearbook publications for 2003 and 2007. The equation q1(t) yield the government strategies and the equation r1(t) the representative firms strategy.

This monograph, a sequel to the author's highly successful i°A Mathematical Treatment of Economic Cooperation and Competition Among Nations: With Nigeria, USA, UK, China and Middle East Examplesi± (Academic Press, 2005), extends the study to all member states of the United Nations. It derives the equations of the key economic variables of gross domestic product, interest rate, employment value of capital stock prices (inflation) and cumulative balance of payment. The derivation is based on the differential market principle of supply and demand and on the rational expectation principle. The models are validated using economic time series of each country and MATLAB programs. The emerging dynamics is a differential game of pursuit which is converted to a hereditary control system for a single nation. Following the same method we derive the full hereditary economic model of all members of the United Nations joined together by external trade, investment capital flow consumption employment and governments' economic intervention ia all mirrored by interacting gross domestic products. The system is validated with IMF and World Bank data. Studies are made on how to arrest economic recession and depression and promote economic growth and prosperity. Diffusion of wealth is also touched upon. Policies regarding economic stimulus, how to dam the decay of capital flow, as well as conditions to promote full employment are discussed. Examination is made to test for the global systems controllability ia the possibility of steering any current bad economic state to a state of growth of its GDP, low interest rate, full employment, good value of capital stock, low inflation and a positive value of the cumulative balance of payments. The book prescribes verifiable broad policies for all nations together to promote prosperity, diffusion of wealth and longevity. The book comes with full programs, output and identified equations which can be downloaded from the publishers' website. Detailed information is available inside the book. The book is a most valuable source of reference for world leaders, central bank directors, graduates and academic researchers in applied mathematics and applied economics concerned with the current problems and growth of the global economy.

In this report, we derive a nonlinear functional differential equation of neutral type as the dynamics of the growth of capital stock of n firms linked up in solidarity in a region. These equations are generalizations of ordinary differential systems studied by Takayama [21, p. 685-706], Arrow [2, p. 184], Knowles [17, p. 3], and Isaacs [13], and delay systems studied by Kalecki [14], Allen [1, p. 254], and Cooke and Yorke [9]. A neutral version (in which there is delay in the derivative) is implicit in the suggestion by Arrow [2, p. 184]. Given an initial capital endowment @f and an initial investment r, we investigate what conditions on the systems parameters and on the external interventions will guarantee that @f can be steered to some fixed target in some finite time. The controls, u, (m-vectors) are investments(if 0=< u"j(t)=< 1) and consumption (if -1 =< u"j(t) =< 0). From the controllability theorems proved for the hereditary systems, we derive universal laws for the control of the growth of capital stock. These laws provide broad, startling, but obvious policy prescription for the growth of companies and for national economies. For example, it describes how big external or governmental interventions on the firms (in the form of taxation or subsidy) must be, relative to the firms' investment and consumption to ensure the growth of capital stock to the given target. It also states when this growth is impossible. We use fixed point theorems and arguments from the theory of games of pursuit [10] to establish constrained controllability of our dynamics. Problems of optimality and interpretations are pursued in a subsequent communication.

This research resolves some controllability questions of general nonlinear delay systems in function space in which earlier solutions of the optimal problem have been shown to exist. By using a powerful open mapping theorem and Gateaux differentiability, we avoid the difficult requirement of Frechet differentiability of an operator F: Wp(1) × Lp → Wp(1) which is associated with the nonlinear system. Properties of the linearized system help ensure the nonlinear system's controllability.

In abstract spaces, we consider certain constrained controllability and approximate controllability properties of a nonlinear system that can be deduced from various controllability properties of its associated linear system. Several examples involving partial differential operators and functional delay operators are given to illustrate the theory.

Sharp, easy to test conditions for exact controllability of nonlinear systems of neutral type in the state space W/sub 2//sup (1)/ are formulated. W/sub 2//sup (1)/ is the Sobolev space of absolutely continuous functions x whose time derivatives are square integrable on the interval (-h,0). Also explored are conditions which guarantee that a null-controllable system is also exactly controllable. Criteria for null controllability of linear systems are given. >

Sufficient conditions are developed for the null-controllability of the nonlinear delay process (1) x (t) = L(t, x t ) + B(t) u(t) + f(t, x t , u(t)) when the values of the control functions u lie in an m-dimensional unit cube Cm of Em. Conditions are placed on f which guarantee that if the uncontrolled system x (t) = L(t, x t ) is uniformly asymptotically stable and if the linear control system x(t) = L(t, xt) + B(t) u(t) is proper, then (1) is null-controllable.

A control process is globally finite time null controllable if it is globally asymptotically stable and locally controllable to the origin. Sufficient conditions are stated for the system \[\dot x = f(t,x,u)\quad {\text{in }}C^1 (R \times R^n \times R^m )\] to be globally finite time null controllable. The conditions are stated in terms of the Jacobian of f and the controllability of a related linear equation.

A new concept of disconjugacy and oscillatoriness is proposed for systems of first-order equations. It is motivated by a reformulation of a problem in time-optimal linear control theory having to do with the parametric description of reachability sets and their frontiers.

Consider the nonlinear neutral functional differential inclusion (i) $$(d/dt)D(t, x_t ) \in R(t, x_t )$$ , whereD is a continuous operator onIXC, linear inx t , indeed of the form (4) below, with kernelD(t, ·)={0}, and atomic at 0, andR is nonempty, closed, and convex. Here,I≡[t 0,t I ] andC=C([-h,0],E n ). In (i), the derivative is specified in terms of the state at timet as well as the state and the derivative of the state for values oft precedingt. We use the Fan fixed-point theorem to prove the existence of a solution of (i) which satisfies two-point boundary values $$x_\omega = \phi _0 ,x_{t_1 } = \phi _1$$ , where φ0, φ1 belong toC. We next apply this existence result to study the exact function space controllability of the neutral functional differential system (ii) $$(d/dt)D(t, x_t ) = f(t, x_t , u), u(t) \in \Omega (t, x_t )$$ . We present sufficient conditions onf and Ω which imply exact controllability between two fixed functions inC.

The paper investigates the equation (1.1) in two cases:(i) p ≡ 0,(ii) p(≠ 0) is bounded. For the case(i) the asymptotic stability (in the large) of the trivial solution x=0 is established, and for the case(ii) a boundedness result is deduced with a bound dependent on the initial conditions. These results on (1.1) are comparable in generality to known results of third and fourth order systems.

The research deals with complete and approximate controllability of the system (∗) dx dy = f(t, x, u) , without control restraints to an arbitrary convex target set. First, some characterizations of complete controllability, to the target of (∗) and a special case of (∗) namely x = A(t)x + k(t, u) ∗∗ are given. As a consequence complete controllability is equivalent to null-controllability. Next certain equations are formulated. These are in the same spirit as J. P. Dauer's “A Controllability Technique for Nonlinear Systems” ( J. Math. Anal. Appl. oo (1972) , 442–451) and are utilized in the main contribution of the paper: Under certain convexity assumption, bounded perturbations of systems which are completely controllable to a fixed target G are completely controllable to G . Without the convexity assumption, but with perturbations satisfying a Lipschitz condition, approximate controllability to G of a perturbed system is equivalent to complete controllability to G of the unperturbed equation.

The paper tackles the time optimal control theory of linear neutral equations. Controllability questions in function space are answered. The existence and forms of an optimal control are established both in function space and in Euclidean space.

The problem of Lurie is posed for systems described by a functional differential equation of neutral type. Sufficient conditions are obtained for absolute stability for the controlled system if it is assumed that the uncontrolled plant equation is uniformly asymptotically stable. Both the direct and indirect control cases are treated.

On etudie le probleme de Lurie pour le systeme dx/dΓ=Ax+bμ, dμ/dx=O(σ), σ=(c,w)-pμ, ou l'operateur A est soit borne soit suppose engendrer un groupe Co fortement continu T(t), t∈(-∞,∞)=E, sur un espace de Hilbert reel X, de produit interieur (., .) et de norme |.|; b, c∈X, μ, p∈E et O: E→E est une fonction non lineaire lipschitzienne uniformement continue

A system is totallyG-controllable if every pointx0 of the state spaceEn can be steered to the targetG in finite time and can be held inG forever afterward. Sufficient conditions are developed for the totalG-controllability of the linear system $$\dot x(t) = A(t)x(t) + B(t)u(t)$$ (a) and its perturbation $$\dot x(t) = A(t)x(t) + B(t)u(t) + F(t,x(t),u(t)),$$ (b) where the targetG is an affine manifold inEn. We state conditions on the perturbation functionF which guarantee that, if (a) is totallyG-controllable, then so is (b). These conditions onF are natural and are obtained by solving a system of nonlinear integral equations by the Leray-Schauder fixed-point theorem.