Adv Syst Sci Appl 2021; 02:8–19 Published online at https://ijassa.ipu.ru. Uncertain Controllability and Observability of an Optimal Control Model Tolulope Latunde1*, Adam A. Ishaq2, Adedayo F. Adedotun3, Joel O. Ajinuhi4, Olumuyiwa J. Peter5 1Department of Mathematics, Federal University Oye-Ekiti, Oye-Ekiti, Nigeria 2Department of Physical Sciences, Al-Hikmah University, Ilorin, Nigeria 3Department of Mathematical Sciences, Olabisi Onabanjo University, Ago-Iwoye, Nigeria 4Department of Mathematics, Federal University of Technology, Minna, Nigeria 5 Department of Mathematics, University of Ilorin, Ilorin, Nigeria Abstract: The interest of this paper is to examine the controllability and observability of a control system in the configuration state-space of an uncertain optimal control system. The control system is designed based on the realization of capital asset values where a special case of asset management is modelled and optimized. Thus some necessary and sufficient conditions of the controllability and observability of the deterministic systems and the corresponding uncertain systems for the case of the uncertain optimal control system with application in capital asset management are considered. Keywords: controllability, observability, uncertain systems, optimal control, capital asset management 1. INTRODUCTION Controllability and observability are important properties in control systems. They represent the ability to move a system around its entire configuration space using certain manipulations. The controllability and observability of a system are mathematical duals that play important roles in control problems such as optimal control. These have played important roles in control theories such as in [1–5]. Recently, researchers such as [6–8] and a host of others have been considering controllability and observability problems in dynamic systems. However, most works done in this area have concentrated on the deterministic and stochastic controllability and observability problems. In this work, uncertain controllability and observability of dynamic systems are carried out by formulating a capital asset management control problem for the uncertain dynamic system such that the uncertain dynamic system is limited to systems involving uncertain processes. The choice of uncertainty theory over the conventional probability theory exists when the sample size is small to estimate a probability distribution and degree beliefs are ascertained from experts to work in place of frequency since human beings always over-weigh unlikely events. Here, the general controllability and observability for the uncertain system in uncertainty theory are presented based on Klamka and Mahmudov works. ∗Corresponding author: tolulope.latunde@fuoye.edu.ng UNCERTAIN CONTROLLABILITY AND OBSERVABILITY OF AN OPTIMAL CONTROL MODEL 9 2. PRELIMINARIES Uncertainty theory is a branch of mathematics for modelling belief degrees. The theory is based on some concepts which may be referred to [9]. For easy interpretation, some of the concepts are given. Let Γ be a nonempty set and L be a σ- algebra over Γ such that (Γ, L) is a measurable space. Each element Λ ∈ L is called an event. Definition 2.1: A set function M defined on the σ-algebra over L is called an uncertain measure if it satisfies the following axioms: Axiom 1. (Normality Axiom): M{Λ} = 1 for the universal set Γ. Axiom 2. (Duality Axiom): M{Λ} + M{Λc} = 1 for any event Λ. Axiom 3. (Subadditivity Axiom): For every countable sequence of events, Λ1,Λ2, . . . , we have M { ∞⋃ i=1 Λi } ≤ ∞∑ i=1 M{Λi}. Axiom 4. (Product Axiom): Let (Γk, Lk,Mk) be uncertainty spaces for k = 1, 2, . . . The product uncertain measure M is an uncertain measure satisfying M { ∞∏ k=1 Λk } = min 1≤k≤∞ Mk{Λk}, where Λk are arbitrarily chosen events from Lk for k = 1, 2, · · · , respectively; see [9]. Definition 2.2: Let (Γ, L,M) be an uncertainty space and let T be a totally ordered set. An uncertain process is a function Xt(γ) from T × (Γ, L,M) to the set of real numbers such that {Xt ∈ B} is an event for any Borel set B of real numbers at each time t; see [10]. Definition 2.3: An uncertain process Cσ is said to be a Liu process if (i) C0 = 0 and almost all sample paths are Lipschitz continuous, (ii) Cσ has stationary and independent increments, (iii) every increment Cs+σ − Cs is a normal uncertain variable with expected value 0 and variance σ2. The uncertainty distribution of Cσ is Φσ(x) = [ 1 + exp ( −πx√ 3σ )]−1 , x ∈ R, (2.1) and the inverse distribution is Φ−1 σ (y) = σ √ 3 π ln y 1− y , y ∈ R; (2.2) see [11]. Definition 2.4: Let ξ be an uncertain variable. Then the expected value of ξ is defined by E[ξ] = ∫ +∞ 0 M{ξ ≥ x}dx− ∫ 0 −∞ M{ξ ≤ x}dx provided that at least one of the two integrals is finite; see [9]. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 10 T. LATUNDE, A.A. ISHAQ, A.F. ADEDOTUN, J.O. AJINUHI, O.J. PETER Definition 2.5: An uncertain process Xt is said to have independent increments if Xt1 −Xt0 , Xt2 −Xt1 , · · · , Xtk −Xtk−1 are independent uncertain variables, where t0 < t1 < · · · < tk. That is, an independent increment process means that its increments are independent uncertain variables whenever the time intervals do not overlap. It is noted that the increments are also independent of the initial state; see [10]. Definition 2.6: Suppose Ct is a canonical Liu process, and f and g are two functions. Then dXt = f(t,Xt)dt+ g(t,Xt)dCt is called an uncertain differential equation. A solution is a Liu process Xt that satisfies (2.1) and (2.2) identically in t; see [10]. Definition 2.7: Let Xt be an uncertain process. Then for each γ ∈ Γ, the function Xt(γ)is called a sample path of Xt; see [10]. Definition 2.8: An uncertain process Xt is said to be sample-continuous if almost all sample paths are continuous functions with respect to time t; see [9]. Definition 2.9: Uncertainty Distribution of Solution. Let α be a number with 0 < α < 1. An uncertain differential equation dX(t) = f(t,X(t))dt+ g(t,X(t))dC(t) is said to have an α-path X(t)α if it solves the corresponding ordinary differential equation dX(t)α = f(t,X(t)α)dt+ |g(t,X(t))|Φ−1(α)dt, where Φ−1(α) is the inverse uncertainty distribution of a standard normal uncertain variable, that is, Φ−1(α) = √ 3 π ln α 1− α , α ∈ R; see [11]. 3. THE ASSET MANAGEMENT MODEL Asset management problem is mainly based on decision making and the understanding of probable asset degradation and trading-off capital investments, maintenance costs, risks and other uncertainties to optimize decisions made by investors. However, it is assumed that an individual invests his/her wealth in a capital asset, A(t), of a large business for a time, t, ranging from t0 to tn. Supposing he/she starts with a known initial net worthX0(t). At the time t, what fraction of his/her net worth, ψ, must he/she choose to utilize on the capital asset and what fraction of his net worth, τ , must he/she choose to be incurred on the liability of the business such that the expected present value of the utility of asset, J(X), is maximized. Table 3.1 represents the definition of the formulated model’s parameters. A dynamic optimization model of the expected present value of assets over a given life cycle based on Uncertainty theory is herein presented following the study of portfolio Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) UNCERTAIN CONTROLLABILITY AND OBSERVABILITY OF AN OPTIMAL CONTROL MODEL 11 Table 3.1. Definition of Parameters to the model. Parameter Description X(t) Net worth at time t (state variable) τ(t) Liability ratio (control) at time t, τ ∈ R σr(t) Diffusion volatility of liability (with variance σ2 r per unit time) ψ(t) Capital asset ratio at time t (control) ψ ∈ R σb(t) Diffusion volatility of asset (with variance σ2 b per unit time) κ(t) Capital gain on asset due to inflation at time t σp(t) Diffusion volatility on asset price (with variance σ2 p per unit time) β(t) Mean rate of return on the asset at time t ω(t) Mean interest rate of liability at time t C(t) Liu canonical process at time t µ(t) Consumption level at time t j(t) Tax ratio at time t g(t) Depreciation ratio at time t h(t) Asset supplies ratio at time t η subjective discount rate, e.g., A η+1 = Present value λ degree of relative risk, where (1− λ) is the risk aversion U Utility function selection by [12]. It is assumed that the goal of the asset management is to choose the optimal utilization and asset allocation policies for maximizing a value function that discounts exponentially future uncertain values of Hyperbolic Absolute Risk Aversion (HARA) utility function over a given time horizon with the net worth of tangible assets as the state variable. The risky asset is assumed to earn an uncertain return and an uncertain gain with the mean rate of return and capital gain. Furthermore, we express the change in liability as the sum of liability service with an assumption of uncertainty, consumptions, investment and net foreign supply, less taxation, depreciation and revenue over a period of time; see [13]. Thus, we have J(X) = max ψ EC  tn∫ t0 1 λ e−ηt(ψX(t))λdt  subject to dX(t) = [(κ+ β)ψ − (ω(ψ − 1) + µ+ h− j − g)]X(t)dt +[ψσp + ψσb − σr(ψ − 1)]X(t)dC(t) . This model has been solved, characterised, analysed and applied to some real-life situations of sustainable finance; see [14–17]. 3.1. Optimality of the Solution It is important to derive the optimal solutions of the capital asset management problem as it would help in selecting the best available values. The following are utilised in deriving the optimality of the proposed model. Definition 3.1: (Principle of Optimality) [19]: For any (t, x) ∈ [0, T )× R and ∆t > 0 with t+ ∆t < T , we have J(t, x) = sup D E [∫ t+∆t t f(Xs, D, S)ds+ J(t+ ∆t, x+ ∆Xt) ] , Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 12 T. LATUNDE, A.A. ISHAQ, A.F. ADEDOTUN, J.O. AJINUHI, O.J. PETER where x+ ∆Xt = Xt+∆t. Theorem 3.1: (Equation of Optimality, [18]) Let J(t, x) be twice differentiable on [0, T )× R, Then we have −Jt(t, x) = sup D [f(x,D, t)) + Jx(t, x)V (x,D)] , where Jt(t, x) and Jx(t, x) are the partial derivatives of the function J(t, x) in t and X respectively. Proof See ( [19], pp. 15-16) 3.2. Optimal control of the model The above equation of optimality is applied to the uncertain optimal control problem to evaluate the optimal controls analytically. Applying equation (3.1), we obtain −Jt = max ψ { 1 λ e−ηt(ψX)λ − ψ(κ+ β)XJX + (µ+ j + g + h− (ψ − 1)ω)XJX } = max ψ H where H stands for terms in the braces (condition the optimal ψ satisfies), ∂H ∂ψ = 0, ∂H ∂ψ = e−ηt(ψX)λ−1X − (κ+ β − ω)XJx = 0, ψ = 1 X [ (ω − κ− β)JXe ηt ] 1 λ−1 . Hence, by solving the above equations, we obtained the optimal ratio of the net worth in capital assets as ψ∗ = (µ+ j + g + ω − h)λ− η (1− λ)(κ+ β − ω) . However, the optimal liability ratio, τ ∗ can also be obtained as a control to the system. Since τ = ψ − 1 τ ∗ = [ (µ+ j + g + ω − h)λ− η (1− λ)(κ+ β − ω) ] − 1 or τ ∗ = (µ+ j + g − h)λ− (1− λ)(κ+ β) + ω − η (1− λ)(κ+ β − ω) . 3.3. Solution to the Model Here, the analytical and numerical solutions are derived. For the analytic solution, the required problem under consideration is J(ψ) = min ψ EC  tn∫ t0 1 λ e−ηt(ψX(t))λdt  Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) UNCERTAIN CONTROLLABILITY AND OBSERVABILITY OF AN OPTIMAL CONTROL MODEL 13 subject to dX(t) = [(κ+ β)ψ − (ω(ψ − 1) + µ+ h− j − g)]X(t)dt +[ψσp + ψσb − σr(ψ − 1)]X(t)dC(t) with α-path equation dX(t)α = [(κ+ β)ψ − (ω(ψ − 1) + µ+ h− j − g)]X(t)αdt +|[ψσp + ψσb − σr(ψ − 1)]X(t)α|Φ−1(α)dt. The analytical solution to the constraint is X(t) = X0 exp ([(κ+ β)ψ − (ω(ψ − 1) + µ+ h− j − g)]t +[ψσp + ψσb − σr(ψ − 1)]C(t)) and its inverse uncertainty distribution is Ψ(t)−1(α) = X0 exp ( [(κ+ β)ψ − (ω(ψ − 1) + µ+ h− j − g)]t + [ψσp + ψσb − σr(ψ − 1)]t √ 3 π ln α 1− α ) . Hence, Ψ(t)−1(α) = E(X(t)α). 3.4. Multifactor Model The multifactor model can be expressed in the following form: J(X) = max ψ EC  tf∫ t0 1 λ e−ηt(Uλ)TX1−λdt  (3.3) subject to dX = FXdt+ UPXdt+ UQXdC(t), (3.4) where X =  X1t X2t ... Xnt  , U =  ψ1 0 · · · 0 0 ψ2 · · · 0 ... . . . ... . . . 0 0 · · · ψn  , F =  µ1 + h1 − j1 − g1 − ω1 0 · · · 0 0 µ2 + h2 − j2 − g2 − ω2 · · · 0 ... . . . . . . ... 0 0 · · · µn + hn − jn − gn − ωm  , P =  κ1 + β1 − ω1 0 · · · 0 0 κ2 + β2 − ω2 · · · 0 ... . . . . . . ... 0 0 · · · κn + βn − ωn  , Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 14 T. LATUNDE, A.A. ISHAQ, A.F. ADEDOTUN, J.O. AJINUHI, O.J. PETER Q =  σ1p + σ1b − σ1r + 1 0 · · · 0 0 σ2p + σ2b − σ2r + 1 · · · 0 ... . . . . . . ... 0 0 · · · σnp + σnb − σnr + 1  . See, for example, [18]. Equations (3.3) and (3.4), which are the model of risky capital assets, is an uncertain optimal control system whereby X(t) is the state, U is the control, F is n× n dimensional constant matrix while P and Q are n×m dimensional constant matrices, C(t) is the Liu process and J(X) is the objective functional. 4. CONTROLLABILITY AND OBSERVABILITY OF THE UNCERTAIN SYSTEM Let (Γ, L,M) be a complete uncertainty space with uncertain measureM on Γ and a filtration {L(t)|t ∈ [0, T ]} generated by n-dimensional uncertain process {C(t) : 0 ≤ t ≤ T} defined on the uncertainty space (Γ, L,M). Let l2(Γ, L(t),Rn) represent the Hilbert space of all L(t)-measurable square integrable uncertain variables with all values in Rn. Also, let lL2 ([0, T ],Rn) represent the Hilbert space of all square integrable and L(t)-measurable process with the values in Rn. Let X(t) = X(t+ s) for s ∈ [0, T ] represent the segment of the trajectory, that is, X(t) ∈ lL2 ([0, T ], l2(Γ, L(t),Rn)). LetR be a linear operator on the Hilbert space l2(Γ, L(t),Rn) with domainD1(R).R ≥ 0 if 〈Rc, c〉 ≥ 0 for all c ∈ D1(R), R > 0 if 〈Rc, c〉 > 0 for all nonzero c ∈ D1(R) and R is coercive (R− γI ≥ 0) if there exists a γ > 0 such that 〈Rc, c〉 ≥ γ ‖c‖2for all c ∈ D1(R). Now, using the proposed model with the multidimensional constraint of uncertain differential equation dX(t) = FX(t)dt+ UPX(t)dt+ UQX(t)dC(t) (4.5) for t ∈ [0, T ] with the function initial condition X0 ∈ lL2 ([0, T ], l2(Γ, L(t),Rn)), where the state X(t) ∈ l2(Γ, L(t),Rn) and control ψ(t) ∈ Rm = U . U and F are n× n dimensional constant matrix while P and Q are n×m dimensional constant matrices. Thus, suppose the admissible controls U = lL2 ([0, T ],Rm), then for any given initial condition X0 ∈ lL2 ([0, T ], l2(Γ, L(t),Rn)) and any admissible control ψ ∈ U for t ∈ [0, T ], there exists a unique solution X(t;X0, ψ) ∈ l2(Γ, L(t),Rn) of the constraint uncertain differential state equation (4.5); see [20]. 4.1. Controllability Definition 4.1: The uncertain dynamic system (4.5) is said to be relatively exactly controllable on [0, T ] if R(t)(U) = l2(Γ, L(t),Rn). This implies that if all the points in l2(Γ, L(t),Rn) can exactly be reached at time T from any arbitrary initial condition X0 ∈ lL2 ([0, T ], l2(Γ, L(t),Rn)). Definition 4.2: The uncertain dynamic system (4.5) is said to be relatively approximately controllable on [0, T ] if R(t)(U) = l2(Γ, L(t),Rn). Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) UNCERTAIN CONTROLLABILITY AND OBSERVABILITY OF AN OPTIMAL CONTROL MODEL 15 This implies that if all the points in l2(Γ, L(t),Rn) can approximately be reached at time T from any arbitrary initial condition X0 ∈ lL2 ([0, T ], l2(Γ, L(t),Rn)). The relationship between the controllability concepts for the uncertain dynamic system (4.5) and the controllability of the related deterministic dynamic system below dX(t) = FX(t)dt+ UPX(t)dt for t ∈ [0, T ], (4.6) where the admissible control ψ ∈ l2([0, T ],Rm). Firstly, the deterministic system (4.6) is defined according to [1]. Let Qj(t) = FQj−1(t) for j = 1, 2, 3, . . . and t > 0, with the initial condition Q(t) = Q0(0) = J, t = 0, Q(t) = Q0(t) = 0, t 6= 0. For instance, the sequence of the n× n dimensional matrices Qj(t) deduced from the determining equation gives: Q0(0) = P, Q1(0) = FP, Q2(0) = F 2P. These can be written in a general notation as Qj(t;T ) = {Q0(t), Q1(t), Q2(t), . . . , Qj−1(t) fort ∈ [0, T ]}. The following Lemma is given with respect to [1], relating to the controllability of the deterministic system (4.6) in the time interval [0, T ]: Lemma 4.1: The following conditions are equivalent: 1. the deterministic system (4.6) is relatively controllable on [0, T ], 2. the relative controllability matrix B(t) is non-singular, 3. rank Qj(t;T ) = k. Hence, the following lemma is formed with respect to [4, 20, 21] which will be useful in the proof of the uncertain controllability and observability. Lemma 4.2: For every c ∈ l2(Γ, L(t),Rn), ∃ a process q ∈ lL2 ,Rn×n such that the controllability operator is expressed as: G(t)c = B(t)Ec+ ∫ T 0 B(t)(s)q(s)dC(s). Lemma 4.3: The uncertain system (4.5) is relatively controllable on [0, T ] if and only if one of the following conditions holds true: 1. E 〈G(t)c, c〉 ≥ γE ‖c‖2 for some γ > 0 and all c ∈ l2(Γ, L(t),Rn), 2. R(λ1, G(t)) converges as λ1 → 0+ in the uniform operator topology, Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 16 T. LATUNDE, A.A. ISHAQ, A.F. ADEDOTUN, J.O. AJINUHI, O.J. PETER 3. λ1R(λ1, G(t)) converges to the zero operator as λ1 → 0+ in the uniform operator topology, 4. ker(l(t))∗ = {0} and Im(l(t))∗. Lemma 4.4: The uncertain system (4.5) is approximately controllable on [0, T ] if and only if one of the following conditions holds true: 1. G(t) > 0, 2. λ1R(λ,G(t)) converges as λ1 → 0+ in the strong operator topology, 3. λ1R(λ,G(t)) converges to the zero operator as λ1 → 0+ in the weak operator topology, 4. ker(l(t))∗ = {0}. Theorem 4.1: The following conditions are equivalent: 1. The deterministic system (4.6) is relatively controllable on [0, T ], 2. The uncertain system (4.5) is relatively exactly controllable on [0, T ], 3. The uncertain system (4.5) is relatively approximately controllable on [0, T ]. Proof Condition (i) implies condition (ii). Suppose the deterministic system (4.6) is relatively controllable on [0, T ], then the relative controllability matrix B(t)(s) is invertible and strictly positive definite for all s ∈ [0, T ]. Hence, for some γ > 0, 〈B(t)(s)X,X〉 ≥ γ ‖X‖2 for all s ∈ [0, T ] and X ∈ Rn. In order to prove the relative exact controllability of the uncertain system (4.5) on [0, T ], the relationship between the controllability operator G(t) and the controllability matrix B(t) given in Lemma 4.2 as E 〈G(t)c, c〉 to be expressed in terms of 〈G(t)Ec,Ec〉. Firstly, E 〈G(t)c, c〉 = E 〈 B(t)Ec+ ∫ T 0 B(t)(s)q(s)dC(s),Ec+ ∫ T 0 q(s)dC(s) 〉 = 〈B(t), Ec, Ec〉+ E ∫ T 0 〈B(t)(s)q(s), q(s)〉 ds ≥ γ ( ‖Ec‖2 + E ∫ T 0 ‖q(s)‖2 ds ) = γE ‖c‖2 . Thus, in the view of the controllability operator, G(t) ≥ γI, which implies that the relative controllability operator G(t) is strictly positive definite and, the inverse operatorG(t)−1 is bounded, [1]. Hence, the uncertain relative exact controllability of uncertain dynamic system (4.5) on [0, T ] is proved from the relative controllability of deterministic system (4.6) on [0, T ] . Condition (ii) implies condition (iii). Since the state space for the uncertain dynamic system (4.5) is finite-dimensional, which implies that the exact and approximate controllability coincide; see [1]. Hence, it is easy to conclude that the uncertain system (4.5) is relatively approximately controllable on [0, T ]. Condition (iii) implies condition (i). Suppose the uncertain dynamic system (4.5) is uncertainly relatively approximately Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) UNCERTAIN CONTROLLABILITY AND OBSERVABILITY OF AN OPTIMAL CONTROL MODEL 17 controllable on [0, T ] and its controllability operator is positive definite, that is, G(t) > 0, then applying the resolvent operator λ1R(λ1, G(t)), where λ > 0, [20], gives E ‖λ1R(λ1, G(t))c‖2 → 0. This implies E ‖λ1R(λ1, G(t))c‖2 = ‖λ1R(λ1, B(t))Ec‖2 + E ∫ T 0 ‖Λ1R(λ1, B(t)(s))q(s)‖2 ds→ 0. Therefore, E ∫ T 0 ‖Λ1R(λ1, B(t)(s))q(s)‖2 ds→ 0 for all q(s) ∈ lL2 [0, T ],Rn×n, and consequently there exists a subsequence λn such that for every c ∈ l2(Γ, L(t),Rn) we have ‖λnR(λk, B(t)(s))c‖ → 0 almost everywhere on [0, T ]. Thus, the property holds for all 0 ≤ s < T because of the continuity of R(λ,B(t)(s)). This implies that the deterministic system (4.6) is relatively approximately controllable on [0, T ]. However, considering the state space for the deterministic system (4.6) is finite- dimensional, that is, exact and approximate controllabilities coincides, [1]. Therefore, it is concluded that the deterministic system (4.6) is relatively controllable on [0, T ]. 4.2. Observability Here, the dual concepts of observability for the uncertain dynamic system (4.5) are treated. Suppose F and J generate C0-semigroup S1 and S2 on the Hilbert space l2(Γ, L(t),Rn), that is, F and J generate a continuous representation of semigroup S. Then the dual of the uncertain system (4.5) is: dX(t) = [F ∗ + J∗ +W ∗ 1 +W ∗ 2 ]X(t)dt+ UQX(t)dC(t). Thus, the following concepts are described: • The observability map of the uncertain system (4.5) on [0, T ] is expressed as the linear operator O(t) : l2(Γ, L(t),Rn)→ lL2 ([0, T ]) which is defined by O(t)c = (W1 +W2)(S1 + S2)(T − s)E{c|L}. • The observability Gramian of the uncertain system (4.5) on [0, T ] is defined by: Θ = (O(t))∗O(t). Definition 4.3: The uncertain system (4.5) is said to be relatively observable on [0, T ] if the operator O(t) is injective and its inverse bounded on the range O(t). The means that the initial state can be uniquely and continuously constructed from outputs in lL2 (Γ, L(t),Rn). Definition 4.4: The uncertain system (4.5) is said to be approximately observable on [0, T ] if O(t) = {0}, that is, the initial state uniquely depends on the knowledge of the result in lL2 (Γ, L(t),Rn). Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 18 T. LATUNDE, A.A. ISHAQ, A.F. ADEDOTUN, J.O. AJINUHI, O.J. PETER Theorem 4.2: For the uncertain dynamic system (4.6), the following duality results hold true: 1. The uncertain system (4.5) is relatively observable on [0, T ] if and only if the dual system (4.2) is relatively controllable on [0, T ]. 2. The uncertain system (4.5) is approximately controllable on [0, T ] if and only if the dual system is approximately controllable on [0, T ]. Proof Since F and J generate a Co-semigroup S1(t) and S2(t) respectively on lL2 (Γ, L(t),Rn), then F ∗ and J∗ generate the Co-semigroup S∗1(t) and S∗2(t) respectively. Also, since O(t) ∈ lL2 ([0, T ], l2(Γ, L(t),Rn)), (O(t))∗q = ∫ T 0 (S∗1 + S∗2)(T − s)(W ∗ 1 +W ∗ 2 )q(s)ds. Thus, the range of O(t) implies that of the controllability operator of the dual system (4.2). Let the controllability of the dual system (4.2) be represented by N(t), then (O(t))∗ = N(t), (N(t))∗ = O(t). 1. Suppose the deterministic system (4.5) is relatively observable, there exists an inverse (O(t))−1 on the range of O(t). Then,∥∥(O(t))−1q ∥∥ ≤ l ‖q‖ for all q ∈ lL2 ([0, T ],Rn×n) and l > 0. Hence, ‖c‖ = ∥∥(O(t))−1O(t)c ∥∥ ≤ l ∥∥OT c ∥∥ = l ‖(N(t))∗c‖ . Therefore, the relative controllability of the uncertain system (4.5) follows from Lemma 4.3 as thus. Suppose that the uncertain system is relatively controllable, then (N(t))∗ is injective and has a closed range. This implies that from (N0)∗ = O(t), O(t) is injective and has a closed range. Thus, by the Closed Graph Theorem, the inverse of O(t) is bounded on the range of O(t). 2. However, by definition, the deterministic system (4.6) is approximately observable if and only if kerO(t) = ker(N(t))∗ = {0}. Therefore by Lemma 4.4, ker(N(t))∗ = {0} if and only if the uncertain system (4.5) is approximately controllable. Hence the proof of the equivalence. 5. CONCLUSION The necessary and sufficient conditions for the uncertain controllability and observability of a finite-dimensional uncertain dynamic control system have been established and proved. Thus, the effectiveness of each input and output in the general operation of the control system can be determined. The application of this work is to measure the controllability and observability degree of the system by using certain admissible input factors to determine how the system can move around its configuration space. Subsequently, we can observe the analytic and numerical solutions to the multifactor system. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) UNCERTAIN CONTROLLABILITY AND OBSERVABILITY OF AN OPTIMAL CONTROL MODEL 19 REFERENCES 1. Klamka, J. (1991) Controllability of Dynamical Systems. Dordrecht: Kluwer Academic, 1991. 2. Klamka, J. (1993) Controllability of Dynamical Systems - A survey. Archives of Control Sciences, 2(3/4), 281–307. 3. Klamka, J. (2007) Stochastic controllability of linear systems with state delays. International Journal of Applied Mathematics and Computer Science, 17(1), 5–13. 4. Mahmudov, N.I.& Denker A. (2000) On controllability of linear stochastic systems. International Journal of Control, 73(2), 144–151. 5. Mahmudov, N.I. (2003) Controllability and observability of linear stochastic systems in Hilbert spaces. Progress in Probability, 53, 151–167. 6. Guo, T.L. (2012) Controllability and observability of impulsive fractional linear time- invariant system. Computer and Mathematics with Applications, 64 3171–3182. 7. Wang, L & Chen, Y. (2017) Physical controllability of complex networks. Scientific Reports, 7, 1–14. 8. Leitold, D., Vathy-Fogarassy, A. & Abonyi, J. (2017) Controllability and observability in complex networks - the effect of connection types. Scientific Reports, 7(151), 1–9. 9. Liu, B. (2016) Uncertain theory (5th ed.), Uncertainty Theory Laboratory, China. 10. Liu, B. (2008) Fuzzy process, hybrid process and uncertain process. Journal of Uncertain Systems 2(1), 3–16. 11. Yao, K. & Chen, X.A. (2013) numerical method for solving uncertain differential equations. Journal of Intelligent and Fuzzy Systems, 25(3), 825–832. 12. Merton, R.C. (1971) Optimal consumption and portfolio rules in a continuous time model. Journal of Economic Theory, 42 (3), 373–413. 13. Latunde, T. & Bamigbola O.M. (2016) Uncertain optimal control model for management of net risky capital asset. IOSR Journal of Mathematics, 12(3), 22–30. 14. Latunde, T. & Bamigbola, O.M. (2016) On existence and uniqueness of solution to a special case of asset management. Journal of the Nigeria Association of Mathematical Physics (NAMP), 35 269–274. 15. Latunde, T. & Bamigbola, O.M. (2016) On the stability analysis of uncertain optimal control model of management of net risky capital asset. International Journal of Applied Sciences and Mathematical Theory, 2(1), 29–36. 16. Latunde, T. & Bamigbola O.M. (2018) Parameter estimation and sensitivity analysis of an optimal control model for capital asset management. Advances in Fuzzy Systems, 2018 1–11. 17. Latunde, T. (2019) Optimal values in an uncertain optimal control model with application to capital asset management. Advances in Systems Science and Applications (ASSA), 19(3), 52–64. 18. Latunde, T. (2020) Multifactor modelling in asset management. International Journal of Mathematics in Operational Research, 17(3), 333–352. 19. Zhu, Y. (2010) Uncertain optimal control with application to a portfolio selection model. Cybernetics and Systems: An International Journal, 41, 535–547. 20. Mahmudov, N.I. (2001) Controllability of linear stochastic systems in Hilbert spaces. Journal of Mathematical Analysis and Application, 251, 64–82. 21. Mahmudov, N.I. & Zorlu, S. (2003) Controllability of nonlinear stochastic systems. International Journal of Control, 76(2), 95–104. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) Introduction Preliminaries The asset management model Optimality of the Solution Optimal control of the model Solution to the Model Multifactor Model Controllability and observability of the uncertain system Controllability Observability Conclusion