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Linear Exponential Quadratic Gaussian Covariance Steering
Chiran B. Cherian, Yasemin Isik, Abhishek Halder
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Summary
This paper formulates and analyzes the Linear Exponential Quadratic Gaussian (LEQG) covariance steering problem in continuous time. It presents the optimal controller as a linear state feedback parameterized by a symmetric matrix solving an algebraic equation, generalizing the risk-neutral LQG case. The authors prove the existence and uniqueness of the solution in the neighborhood of the risk-neutral optimal solution for matched noise and input channels.
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Chiran B. Cherian → affiliatedwith → Iowa State University
confidence 99% · Chiran B. Cherian and Abhishek Halder are with the Department of Aerospace Engineering, Iowa State University
Yasemin Isik → affiliatedwith → Iowa State University
confidence 99% · Yasemin Isik is with the Department of Mathematics, Iowa State University
LEQG covariance steering → isageneralizationof → LQG covariance steering
confidence 95% · Problem (2) can be seen as a risk-sensitive generalization of the risk-neutral linear quadratic Gaussian (LQG) covariance steering problem
LEQG covariance steering → fundedby → NSF
confidence 90% · This research was partially supported by NSF awards 2111688, 2450377.
LEQG covariance steering → isinterpretedas → Schrödinger Bridge
confidence 90% · The solution for this problem can be seen as a risk-sensitive Schrödinger bridge between Gaussian endpoints
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Abstract
Abstract:We formulate and analyze the linear exponential quadratic Gaussian (LEQG) covariance steering problem in continuous time over a given deadline (finite time horizon). The solution for this problem can be seen as a risk-sensitive Schrödinger bridge between Gaussian endpoints in the linear quadratic setting. Unlike the risk-neutral case, the LEQG covariance steering controller--still a linear state feedback--can no longer be written in closed form. We show that the optimal controller is parameterized by a symmetric matrix solving an algebraic equation that encodes the implicit dependence on the risk-sensitivity parameter. We explain how the structure of this optimal controller significantly generalizes the existing results for the risk-neutral case. Building on these results, for the matched noise and input channel case, we prove the existence-uniqueness of solution for the LEQG covariance steering problem in the neighborhood of the known risk-neutral optimal solution. We give an illustrative numerical example.
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- Source: https://arxiv.org/abs/2609.12463v1
- Canonical: https://arxiv.org/abs/2609.12463v1
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Linear Exponential Quadratic Gaussian Covariance Steering Chiran B. Cherian Yasemin Isik Abhishek Halder †thanks: Chiran B. Cherian and Abhishek Halder are with the Department of Aerospace Engineering, Iowa State University, Ames, IA 50011, USA, cbckbc,ahalder@iastate.edu. Yasemin Isik is with the Department of Mathematics, Iowa State University, Ames, IA 50011, USA, yisik@iastate.edu. †thanks: This research was partially supported by NSF awards 2111688, 2450377. Abstract We formulate and analyze the linear exponential quadratic Gaussian (LEQG) covariance steering problem in continuous time over a given deadline (finite time horizon). The solution for this problem can be seen as a risk-sensitive Schrödinger bridge between Gaussian endpoints in the linear quadratic setting. Unlike the risk-neutral case, the LEQG covariance steering controller–still a linear state feedback–can no longer be written in closed form. We show that the optimal controller is parameterized by a symmetric matrix solving an algebraic equation that encodes the implicit dependence on the risk-sensitivity parameter. We explain how the structure of this optimal controller significantly generalizes the existing results for the risk-neutral case. Building on these results, for the matched noise and input channel case, we prove the existence-uniqueness of solution for the LEQG covariance steering problem in the neighborhood of the known risk-neutral optimal solution. We give an illustrative numerical example. I Introduction For the standard integral quadratic cost C:=∫0112(xt⊤Qtxt+ut⊤Rtut)dt,Qt⪰0,Rt≻0, C:=\! _0^1\! 12 (x_t Q_tx_t+u_t R_tu_t ) t,\;Q_t 0,\;R_t 0, (1) and fixed finite θ∈ℝθ , we consider the linear exponential quadratic Gaussian (LEQG) covariance steering problem: infγ∈Γ _γ∈ 1θlog[exp(θC)] 1θ [ (θ C ) ] (2a) subject to dxt=(Atxt+Btut)dt+Ftdwt, x_t= (A_tx_t+B_tu_t ) t+F_t\, w_t, (2b) x0∼ρ0:=(0,Σ0),x1∼ρ1:=(0,Σ1), x_0 _0:=N(0, _0), x_1 _1:=N(0, _1), (2c) in continuous time t∈[0,1]t∈[0,1], where the state xt∈ℝnx_t ^n, the control input ut∈ℝmu_t ^m, and the standard Wiener process wt∈ℝpw_t ^p. Problem (2) seeks to design a Markovian feedback controller γ:[0,1]×ℝn↦ℝmγ:[0,1]×R^n ^m, i.e., ut=γ(t,x)u_t=γ(t,x), such that the optimally controlled state is steered between the centered Gaussians ρ0,ρ1 _0, _1 with given covariances Σ0,Σ1 _0, _1 over the given time horizon [0,1][0,1]. The set of feasible control policies is Γ:=γ:[0,1]×ℝn↦ℝm∣∫01∥γ(t,x)∥22dt<∞, :=\γ:[0,1]×R^n ^m _0^1\|γ(t,x)\|_2^2 t<∞\, where the expectation operator [⋅]E [· ] is taken w.r.t. the controlled state under the chosen policy γ. For a given γ∈Γγ∈ , the [⋅]E [· ] in (2a) is w.r.t. the law of the controlled state. There is no loss of generality in considering the endpoint Gaussians in (2c) to be centered because the solution for the nonzero-mean case is obtained by adding a time-varying drift to the optimal controller for problem (2). This time-varying drift depends on the solution of problem (2) but not the other way around; see [1, Remark 9] mutatis mutandis for θ≠0θ≠ 0. Throughout, we make the following assumptions. Assumptions. A1. The trajectory tuple (At,Bt,Ft)(A_t,B_t,F_t) is bounded and continuous in t∈[0,1]t∈[0,1], and the pair (At,Bt)(A_t,B_t) is uniformly controllable over the time horizon [0,1][0,1]. A2. The matricial trajectories Qt⪰0Q_t 0 and Rt≻0R_t 0 are bounded and continuous w.r.t. t∈[0,1]t∈[0,1]. A3. The given covariance matrices Σ0,Σ1≻0 _0, _1 0. Problem (2) can be seen as a risk-sensitive generalization of the risk-neutral linear quadratic Gaussian (LQG) covariance steering problem–the latter differs from (2) only in that the objective then is replaced by the average cost [C]E[C]. The constant θ∈ℝθ is called the risk-sensitivity parameter. Notice that log[exp(θC)] [ (θ C ) ] is the logarithmic moment generation function (log-MGF) of the random variable C. By L’Hôpital’s rule, limθ→01θlog[eθC]=[C] _θ→ 0 1θ [e^θ C]=E[C], i.e., the limit θ→0θ→ 0 corresponds to the risk-neutral LQG objective. When θ>0θ>0, the LEQG objective becomes risk-averse. When θ<0θ<0, the objective becomes risk-seeking. While the LQG problem minimizes the average cost [C]E[C], the LEQG objective is such that the realizations of the (random) cost C that are above average, are penalized more than the realizations of C that are below average. So minimizing 1θlog[eθC] 1θ [e^θ C] ensures that for θ>0θ>0 large, a larger C is more aggressively penalized. The motivation is that if the random variable C has a large variance, then we prefer a more conservative controller that might be worse on average, but perform better in the worst case. Related Works. In classical stochastic control, the solution of the LEQG problem with terminal cost is well-known via dynamic programming [2, Sec. VIII], via stochastic maximum principle [3, 4], and via completion-of-square and Radon-Nikodym derivative [5]. Notice, however, that problem (2) has no terminal cost and instead, has endpoint covariance constraints. On the other hand, the LQG covariance steering problem (i.e., with objective [C]E[C]) was first studied in [1, 6, 7]. Specifically, the work in [1] derived the optimal controller for the case Ft=Bt,Qt≡0F_t=B_t,Q_t≡ 0. The developments in [6] considered different input and noise coefficient matrices (i.e., Ft≠BtF_t≠ B_t) and Qt≡0Q_t≡ 0. The work in [7] obtained the solution for Ft=BtF_t=B_t and Qt⪰0Q_t 0. The purpose of our work is to initiate a generalization of these results for the risk-sensitive objective 1θlog[eθC] 1θ [e^θ C]. From a probabilistic viewpoint, problem (2) can be interpreted as a risk-sensitive variant of the Schrödinger bridge problem in the linear quadratic setting [8, 9]: inf(ρ,γ)∈01×Γ1θlog[exp(θC)] _(ρ,γ) _01× 1θ [ (θ C ) ] (3a) subject to∂tρ+∇x⋅(ρ(Atx+Btγ))=12⟨FtFt⊤,∇x2ρ⟩, \;subject to _tρ+ _x· (ρ (A_tx+B_tγ ) )= 12 F_tF_t , _x^2ρ , (3b) where 01P_01 is the collection of probability density function (PDF)-valued curves that are continuous in t∈[0,1]t∈[0,1], and connect the given centered Gaussian endpoints ρ0,ρ1 _0, _1. In (3b), the symbol ⟨⋅,⋅⟩ ·,· denotes the Frobenius inner product, and ∇x⋅,∇x2 _x·, _x^2 denote the Euclidean divergence and Hessian, respectively. In particular, (3b) is the Fokker-Planck or Kolmogorov’s forward PDE describing the evolution of state PDF ρ(t,x)ρ(t,x) under the controlled dynamics. In this letter, we will not develop this connection (i.e., the large deviation viewpoint). Instead, we will focus on the structure and computation of the optimal solution. Contributions. For the LEQG problem (2), we • deduce the optimal solution structure, and explain how this structure departs from the existing results known for the risk-neutral case (Sec. I), • establish existence-uniqueness of the optimal solution in the neighborhood of the risk-neutral optimal solution (Theorem 1 and Sec. I), • implement a numerical procedure that builds on the structural results deduced here, and illustrate the solution via a noisy double integrator example (Sec. IV). Notations and acronyms. For natural number n>1n>1, we use nS^n to denote the set of n×n× n real symmetric matrices, and ++nS^n_++ to denote its subset comprising of positive definite matrices. The notation λi _i denotes the iith eigenvalue, and ⊗ denotes the Kronecker product. We use the acronyms IVP and BVP for initial and boundary value problem, respectively. We abbreviate “state transition matrix” as STM, and “with respect to” as w.r.t. I Main Results I-A Necessary Conditions for Optimality Proposition 1. Consider problem (2) with fixed finite θ∈ℝθ (equivalently problem (3)) and assumptions A1-A3. The optimal controller is γopt(t,x)=−Rt−1Bt⊤Πt|θx, _opt(t,x)=-R_t^-1B_t _t θ\>x, (4) where (Ht|θ,Πt|θ) (H_t θ, _t θ ) jointly solve11 1 The subscript t|θt θ is meant to signify the parametric dependence of the time varying matrix on the given risk-sensitivity parameter θ∈ℝθ . the coupled Riccati-Riccati matrix BVP: −H˙t|θ= - H_t θ= At⊤Ht|θ+Ht|θAt+Ht|θBtRt−1Bt⊤Ht|θ−Qt A _tH_t θ+H_t θA_t+H_t θB_tR_t^-1B_t H_t θ-Q_t −(Πt|θ+Ht|θ)(BtRt−1Bt⊤−FtFt⊤)(Πt|θ+Ht|θ) -\! ( _t θ+H_t θ ) (B_tR_t^-1B_t -F_tF _t ) ( _t θ+H_t θ ) −θΠt|θFtFt⊤Πt|θ, - MidnightBlue!10$θ _t θF_tF_t _t θ$, (5a) −Π˙t|θ= - _t θ= At⊤Πt|θ+Πt|θAt+Qt A_t _t θ+ _t θA_t+Q_t −Πt|θ(BtRt−1Bt⊤−θFtFt⊤)Πt|θ, - _t θ (B_tR_t^-1B_t - MidnightBlue!10$θ F_tF_t $ ) _t θ, (5b) Σ0−1= _0^-1= Π0|θ+H0|θ,Σ1−1=Π1|θ+H1|θ. _0 θ+H_0 θ, _1^-1= _1 θ+H_1 θ. (5c) Proof. That the optimal feedback policy is a linear state feedback follows from the standard linear-quadratic theory arguments [2, 5, 1, 7]. In particular, the necessary conditions of optimality for (2) yields an ODE BVP in the covariance state-costate pair (Σt|θ,Πt|θ)∈++n×n( _t θ, _t θ) ^n_++×S^n, given by Σ˙t|θ= _t θ= (At−BtRt−1Bt⊤Πt|θ)Σt|θ (\!A_t-B_tR_t^-1B_t _t θ\! )\! _t θ +Σt|θ(At−BtRt−1Bt⊤Πt|θ)⊤+FtFt⊤, + _t θ (\!A_t-B_tR_t^-1B_t _t θ\! )^\!\! +F_tF_t , (6a) −Π˙t|θ= - _t θ= At⊤Πt|θ+Πt|θAt+Qt A_t _t θ+ _t θA_t+Q_t −Πt|θ(BtRt−1Bt⊤−θFtFt⊤)Πt|θ, - _t θ (B_tR_t^-1B_t -θ F_tF_t ) _t θ, (6b) Σ0|θ=Σ0≻0given,Σ1|θ=Σ1≻0given. -28.90755pt _0 θ= _0 0\;given, _1 θ= _1 0\;given. (6c) The optimal controller (4) follows from the Pontryagin’s minimum principle. Notice that (6a)-(6b) comprise a one-way coupled system of Lyapunov-Riccati ODEs subject to boundary conditions (6c) only in covariance. We use the following change-of-variable Σt|θ∈++n→Ht|θ∈n _t θ ^n_++→ H_t θ ^n proposed in [1]: Ht|θ:=Σt|θ−1−Πt|θ, H_t θ:= _t θ^-1- _t θ, (7) to rewrite the coupled Lyapunov-Riccati BVP (6) in indeterminate pair (Σt|θ,Πt|θ) ( _t θ, _t θ ) as the Riccati-Riccati BVP (5) in indeterminate pair (Ht|θ,Πt|θ) (H_t θ, _t θ ). ∎ In the risk neutral case (θ=0θ=0), the highlighted terms in (5a)-(5b) vanish. If in addition, the noise and input channels coincide, i.e., FtFt⊤=BtRt−1Bt⊤, F_tF _t=B_tR_t^-1B_t , (8) then (5) reduces to the boundary value problem in [7, eq. (2)]. The system of equations (4)-(5) can be seen as a risk-sensitive generalization of the same in prior work [7]. To isolate the complexities due to the noise and input channel mismatch22 2 studied in [6] for covariance steering, and more recently in [10] for distribution steering in general. from that of risk-sensitivity, in the remaining of this letter, we consider the version of (5a)-(5b) together with (8), i.e., −H˙t|θ= - H_t θ= At⊤Ht|θ+Ht|θAt+Ht|θBtRt−1Bt⊤Ht|θ−Qt A _tH_t θ+H_t θA_t+H_t θB_tR_t^-1B_t H_t θ-Q_t −θΠt|θBtRt−1Bt⊤Πt|θ, - MidnightBlue!10$θ _t θB_tR_t^-1B_t _t θ$, (9a) −Π˙t|θ= - _t θ= At⊤Πt|θ+Πt|θAt+Qt A_t _t θ+ _t θA_t+Q_t +(θ−1)Πt|θBtRt−1Bt⊤Πt|θ, + MidnightBlue!10$(θ-1) _t θB_tR_t^-1B_t _t θ$, (9b) and the boundary conditions (5c). As earlier, the highlights in (9) emphasize the terms that are new compared to the risk-neutral setting. In Sec. I-B, we will explain how these terms give rise to new mathematical challenges not seen in the risk-neutral case. Our main contribution is to establish the existence-uniqueness of the solution of the system of equations (9) and (5c), in the neighborhood of the risk-neutral (θ=0θ=0) solution33 3 The existence-uniqueness of the corresponding risk-neutral solution was established in [7, Theorem 1].. To state our main result (Theorem 1), we need to parameterize the solution for (9) and (5c) in terms of certain matrix Y0|θ∈nY_0 θ ^n and the given risk-sensitivity parameter θ∈ℝθ . In Sec. I-B, we detail this (implicit) parametrization. I-B Parametric Solution Structure Our starting point is to write the solutions of the Riccati ODEs (9a) and (9b) in terms of the solutions of the following linear Hamiltonian ODEs: [X^˙t|θY^˙t|θ]= bmatrix X_t θ\\ Y_t θ bmatrix\!\!= [At−BtRt−1Bt⊤−(Qt+θΠt|θBtRt−1Bt⊤Πt|θ)−At⊤]⏟=:M^t|θ[X^t|θY^t|θ], \!\! bmatrixA_t&-B_tR_t^-1B _t\\ -(Q_t+θ _t θB_tR_t^-1B _t _t θ)&-A_t bmatrix_=: M_t θ\!\! bmatrix X_t θ\\ Y_t θ bmatrix, (10a) [X˙t|θY˙t|θ]= bmatrix X_t θ\\ Y_t θ bmatrix\!\!= [At(θ−1)BtRt−1Bt⊤−Qt−At⊤]⏟=:Mt|θ[Xt|θYt|θ]. \! bmatrixA_t&(θ-1)B_tR_t^-1B _t\\ -Q_t&-A_t bmatrix_=:M_t θ bmatrixX_t θ\\ Y_t θ bmatrix. (10b) The solutions of (9a) and (9b) admit the well-known [11, p. 156] linear fractional representations Ht|θ=−(X^t|θ⊤)−1Y^t|θ⊤,Πt|θ=Yt|θXt|θ−1, H_t θ=- ( X_t θ )^-1 Y_t θ , _t θ=Y_t θX_t θ^-1, (11) provided that Xt|θ,X^t|θX_t θ, X_t θ are invertible ∀t∈[0,1]∀ t∈[0,1], θ∈ℝθ . Remark 1. Unlike the risk-neutral case in [7], M^t|θ≠Mt|θ M_t θ≠ M_t θ for θ≠0θ≠ 0. For 0≤s<t≤10≤ s<t≤ 1, let Φ^s→t|θ,Φs→t|θ∈ℝ2n×2n _s→ t θ, _s→ t θ ^2n× 2n denote the STMs from time s to t, associated with the coefficient matrices M^t|θ,Mt|θ M_t θ,M_t θ, respectively, i.e., ∂tΦ^s→t|θ=M^t|θΦ^s→t|θ,Φ^s→s|θ=I, ∂ t _s→ t θ= M_t θ _s→ t θ, _s→ s θ=I, (12a) ∂tΦs→t|θ=Mt|θΦs→t|θ,Φs→s|θ=I. ∂ t _s→ t θ=M_t θ _s→ t θ, _s→ s θ=I. (12b) Expressing the corresponding STMs in terms of their n×n× n blocks, for any t∈[0,1]t∈[0,1], we write [X^t|θY^t|θ]=[Φ^0→t|θ11Φ^0→t|θ12Φ^0→t|θ21Φ^0→t|θ22]⏟=Φ^0→t|θ[X^0|θY^0|θ], bmatrix X_t θ\\ Y_t θ bmatrix\!=\! bmatrix _0→ t θ^11& _0→ t θ^12\\ _0→ t θ^21& _0→ t θ^22 bmatrix_= _0→ t θ\! bmatrix X_0 θ\\ Y_0 θ bmatrix, [Xt|θYt|θ]=[Φ0→t|θ11Φ0→t|θ12Φ0→t|θ21Φ0→t|θ22]⏟=Φ0→t|θ[X0|θY0|θ]. bmatrixX_t θ\\ Y_t θ bmatrix\!=\! bmatrix _0→ t θ^11& _0→ t θ^12\\ _0→ t θ^21& _0→ t θ^22 bmatrix_= _0→ t θ\! bmatrixX_0 θ\\ Y_0 θ bmatrix\!. As in the proof of [7, Theorem 1], without loss of generality, we can set X^0|θ=X0|θ=I X_0 θ=X_0 θ=I since their initial values can be absorbed into Y^0|θ,Y0|θ Y_0 θ,Y_0 θ keeping H0|θ,Π0|θ∈nH_0 θ, _0 θ ^n invariant. Then from (11), both Y^0|θ,Y0|θ Y_0 θ,Y_0 θ are symmetric. In particular, combining (7) and (11), we obtain Y^0|θ=Y0|θ−Σ0−1. Y_0 θ=Y_0 θ- _0^-1. (13) Furthermore, Xt|θ X_t θ =Φ0→t|θ11+Φ0→t|θ12Y0|θ, = _0→ t θ^11+ _0→ t θ^12Y_0 θ, (14a) Yt|θ Y_t θ =Φ0→t|θ21+Φ0→t|θ22Y0|θ, = _0→ t θ^21+ _0→ t θ^22Y_0 θ, (14b) and hence from (11), the matrix Πt|θ∈n _t θ ^n is determined by the choice of Y0|θ∈nY_0 θ ^n. A notable departure from the risk-neutral case is that the matrix M^t|θ M_t θ in (10a) depends explicitly on Πt|θ∈n _t θ ^n, and therefore on Y0|θY_0 θ. Hence, the blocks of Φ^0→t|θ _0→ t θ should be viewed as functions of both θ and Y0|θY_0 θ. In contrast, the blocks of Φ0→t|θ _0→ t θ are functions of θ, but do not depend on Y0|θY_0 θ. Particularizing the above observations for t=1t=1, we find H1|θ=−(X^1|θ⊤)−1Y^1|θ⊤= H_1 θ=- ( X_1 θ )^-1 Y_1 θ = −((Φ^0→1|θ11)⊤+Y^0|θ(Φ^0→1|θ12)⊤)−1((Φ^0→1|θ21)⊤+Y^0|θ(Φ^0→1|θ22)⊤), -\! (\!\! (\! _0→ 1 θ^11 )^\!\! \!\!\!+\! Y_0 θ (\! _0→ 1 θ^12 )^\!\! \! )^\!\!-1\!\!\!\! (\!\! (\! _0→ 1 θ^21\! )^\!\! \!\!\!+\! Y_0 θ\! (\! _0→ 1 θ^22\! )^\!\!\! \! )\!, (15a) Π1|θ=Y1|θX1|θ−1 _1 θ=Y_1 θX_1 θ^-1 =(Φ0→1|θ21+Φ0→1|θ22Y0|θ)(Φ0→1|θ11+Φ0→1|θ12Y0|θ)−1. =\! (\! _0→ 1 θ^21+ _0→ 1 θ^22Y_0 θ\! )\!\! (\! _0→ 1 θ^11+ _0→ 1 θ^12Y_0 θ\! )^\!\!-1\!\!. (15b) Since (13) allows us to write Y^0|θ Y_0 θ in terms of Y0|θY_0 θ, we view (15a)-(15b) as functions of θ and Y0|θY_0 θ. Therefore, using (7), we arrive at an implicit algebraic equation in indeterminate Y0|θY_0 θ given by G(θ,Y0|θ):=Π1|θ+H1|θ−Σ1−1=0. G(θ,Y_0 θ):= _1 θ+H_1 θ- _1^-1=0. (16) If a matrix Y0|θ∈nY_0 θ ^n solving (16) exists and is unique in a suitable sense, then it determines Πt|θ _t θ (via (11)), and thus the optimal control (4). However, an explicit solution of (16) in the form Y0|θ=Y0|θ(θ)Y_0 θ=Y_0 θ(θ) is not available in general. In the risk-neutral (θ=0θ=0) case, M^t|0=Mt|0 M_t 0=M_t 0, so the corresponding STMs match44 4 Xt|0≠X^t|0X_t 0≠ X_t 0 in general, since they solve the same Hamiltonian ODE but with different initial conditions. This makes Lemma 1 in Sec. I nontrivial., i.e., Φ^s→t|0=Φs→t|0 _s→ t 0= _s→ t 0. This STM matching plays a crucial role in the solution derived in [7, Theorem 1]. In that case, combining the terminal boundary condition Σ1−1=Π1|0+H1|0 _1^-1= _1 0+H_1 0 with (15), and using (13), the symplectic identities in [7, Lemma 3] turn (16) into a quadratic equation in unknown Y0|0Y_0 0. In the proof for [7, Theorem 1], the two roots (Y0|0+,Y0|0−Y_0 0^+,Y_0 0^-) of this quadratic equation found via completion-of-square, were Y0|0±=12Σ0−1−(Φ0→112)−1Φ0→111 Y_0 0^±= 12 _0^-1- ( ^12_0→ 1 )^-1 ^11_0→ 1 ±Σ0−12(14I+Σ012(Φ0→112)−1Σ1(Φ0→112)−⊤Σ012)12Σ0−12. ± _0^- 12 ( 14I+ _0 12 ( ^12_0→ 1 )^-1 _1 ( ^12_0→ 1 )^- _0 12 ) 12 _0^- 12. (17) In that work, it was shown that only one of these roots, namely Y0|0−Y_0 0^-, ensures invertibility55 5 Their lack of invertibility for Y0|0+Y_0 0^+ implies finite escape times for Πt|0 _t 0,Ht|0H_t 0 for some t∈[0,1]t∈[0,1], i.e., inadmissible initial condition for (9) at θ=0θ=0. of Xt|0,X^t|0X_t 0, X_t 0 ∀t∈[0,1]∀ t∈[0,1]. By (11), this in turn ensures the unique admissible solution of (9) for θ=0θ=0. In our θ≠0θ≠ 0 case, per Remark 1, Φ^s→t|θ≠Φs→t|θ _s→ t θ≠ _s→ t θ, and the mixed STM terms no longer satisfy the same symplectic identities. Consequently, we have to contend with the implicit algebraic equation (16) in Y0|θY_0 θ. Now we are ready to state our main result. Theorem 1. Consider problem (2) with fixed finite θ∈ℝθ , assumptions A1-A3, and the matched channel condition (8). Let Y0|0−Y_0 0^- denote the unique admissible solution of (16) in the risk-neutral LQG covariance steering problem. Then there exist open sets ℐ⊂ℝI , ⊂nU ^n such that (0,Y0|0−)∈ℐ× (0,Y_0 0^- ) ×U, and a unique 1C^1 mapping h:ℐ→h:I such that h(0)=Y0|0−h(0)=Y_0 0^- and G(θ,h(θ))=0G(θ,h(θ))=0 ∀θ∈ℐ∀θ . In other words, the LEQG problem (2) has a local 1C^1 solution branch66 6 For θ∈ℐθ , both X^t|θ,Xt|θ X_t θ,X_t θ remain invertible ∀t∈[0,1]∀ t∈[0,1] by continuity (determinants are continuous in θ, and by [7, Thm. 1], invertibility holds at θ=0θ=0). through Y0|0−Y_0 0^-. In Section I that follows, we prove Theorem 1 using several auxiliary results (Proposition 2, Lemma 1, Proposition 3) derived therein. I Proofs and Auxiliary Results Our results in this Section are organized and interconnected as in the block diagram below. Proposition 2. Consider G in (16) as a nonlinear map G(θ,Y0|θ):ℐ×↦G(θ,Y_0 θ):I×U where ℐ×I×U and V are bounded open subsets of ℝ×nR×S^n and nS^n, respectively. Then G∈1(ℐ×,)G ^1(I×U;V). Proof. We establish the desired regularity of G w.r.t. the tuple (θ,Y0|θ) (θ,Y_0 θ ) by following the dependence structure of the quantities entering the definition of G: (θ,Y0|θ)⟼Πt|θ(θ,Y0|θ)⟼Ht|θ(θ,Y0|θ)⟼G(θ,Y0|θ). (θ,Y_0 θ )\! \! _t θ (θ,Y_0 θ )\! \!H_t θ (θ,Y_0 θ )\! \!G(θ,Y_0 θ). All matrix spaces involved are finite-dimensional and may therefore be identified with Euclidean spaces. Consider the Riccati ODE (9b) in indeterminate Πt|θ _t θ with initial condition Π0|θ=Y0|θ _0 θ=Y_0 θ, where θ serves as a parameter. The corresponding vector field is continuous in t, and 1C^1 w.r.t. (Πt|θ,θ)( _t θ,θ) since it comprises of continuous and bounded in time At,Bt,QtA_t,B_t,Q_t (Assumptions A1-A2), together with finite sums and products involving Πt|θ _t θ and θ. Therefore, by the smoothness theorem for parameter-dependent ODE IVP [12, Chapter V, Theorem 3.1], Πt|θ _t θ is jointly 1C^1 w.r.t. (θ,Y0|θ)(θ,Y_0 θ). Next, consider the Riccati ODE (9a) in indeterminate Ht|θH_t θ with initial condition (see (7)) H0|θ=Σ0−1−Y0|θH_0 θ= _0^-1-Y_0 θ (which is affine, thus 1C^1 in Y0|θY_0 θ). Since Πt|θ _t θ is jointly 1C^1 w.r.t. (θ,Y0|θ)(θ,Y_0 θ), the vector field of (9a) is continuous in t, and 1C^1 w.r.t. (Ht|θ,θ,Y0|θ)(H_t θ,θ,Y_0 θ). By the same smoothness theorem [12, Chapter V, Theorem 3.1], Ht|θH_t θ is jointly 1C^1 w.r.t. (θ,Y0|θ)(θ,Y_0 θ). Since evaluation at the fixed terminal time t=1t=1 preserves 1C^1-regularity, both Π1|θ,H1|θ _1 θ,H_1 θ are 1C^1 in (θ,Y0|θ)(θ,Y_0 θ). The matrix Σ1−1 _1^-1 is constant. Therefore, G∈1(ℐ×,)G ^1(I×U;V). ∎ Lemma 1. At θ=0θ=0, the partial Jacobian DY0|0G=X1|0−⊤⊗X1|0−⊤−X^1|0−⊤⊗X^1|0−⊤. D_Y_0 0G=X_1 0^- X_1 0^- - X_1 0^- X_1 0^- . (18) Proof. From (16), we have dG(0,Y0|0)=dΠ1|0+dH1|0. G(0,Y_0 0)= _1 0+ H_1 0. (19) Using (15b), dΠ1|0=d(Y1|0X1|0−1)=(dY1|0)X1|0−1−Y1|0X1|0−1(dX1|0)X1|0−1 _1 0= \! (Y_1 0X_1 0^-1\! )\!=( Y_1 0)X_1 0^-1-Y_1 0X_1 0^-1( X_1 0)X_1 0^-1 =(Φ0→1|022dY0|0)X1|0−1−Π1|0(Φ0→1|012dY0|0)X1|0−1 =( ^22_0→ 1 0 Y_0 0)X_1 0^-1- _1 0( ^12_0→ 1 0 Y_0 0)X_1 0^-1 =(Φ0→1|022−Π1|0Φ0→1|012)(dY0|0)X1|0−1. = ( ^22_0→ 1 0- _1 0 ^12_0→ 1 0 ) ( Y_0 0 )X_1 0^-1. (20) Since Π1|0∈n _1 0 ^n, we have Π1|0=Π1|0⊤=X1|0−⊤Y1|0⊤ _1 0= _1 0 =X_1 0^- Y_1 0 . Let A:=Φ0→1|022−Π1|0Φ0→1|012=Φ0→1|022−X1|0−⊤Y1|0⊤Φ0→1|012A:= ^22_0→ 1 0- _1 0 ^12_0→ 1 0= ^22_0→ 1 0-X_1 0^- Y_1 0 ^12_0→ 1 0. Left-multiplying A by X1|0⊤=(Φ0→1|011)⊤+Y0|0(Φ0→1|012)⊤X_1 0 = (\! _0→ 1 0^11\! )^\!\! \!+\!Y_0 0\! (\! _0→ 1 0^12\! )^\! , we obtain X1|0⊤A X_1 0^\! A =(Φ0→1|011)⊤Φ0→1|022+Y0|0(Φ0→1|012)⊤Φ0→1|022 = (\! ^11_0→ 1 0\! )^\! \!\! _0→ 1 0^22+Y_0 0 (\! _0→ 1 0^12\! )^\! _0→ 1 0^22 −(Φ0→1|021)⊤Φ0→1|012−Y0|0(Φ0→1|022)⊤Φ0→1|012 - (\! _0→ 1 0^21\! )^\! \!\! _0→ 1 0^12-Y_0 0 (\! _0→ 1 0^22\! )^\! _0→ 1 0^12 =(Φ0→1|011)⊤Φ0→1|022−(Φ0→1|021)⊤Φ0→1|012=I, = (\! ^11_0→ 1 0\! )^\! \!\! _0→ 1 0^22- (\! _0→ 1 0^21\! )^\! \!\! _0→ 1 0^12=I, where the second equality is due to the symplectic identity [7, eq. (7b)], and the third is due to the symplectic identity [7, eq. (7a)]. Thus77 7 This also shows X1|0X_1 0 is nonsingular. A=X1|0−⊤A=X_1 0^- , which simplifies (21) to dΠ1|0=X1|0−⊤(dY0|0)X1|0−1. _1 0=X_1 0^- ( Y_0 0 )X_1 0^-1. (21) Starting from (15a), similar computation88 8 This also shows that X^1|0 X_1 0 is nonsingular. yields dH1|0=−X^1|0−⊤(dY0|0)X^1|0−1. H_1 0=- X_1 0^- ( Y_0 0 ) X_1 0^-1. (22) Substituting dΠ1|0 _1 0 and dH1|0 H_1 0 from (21) and (22) back into (19), we obtain dG(0,Y0|0)=X1|0−⊤(dY0|0)X1|0−1−X^1|0−⊤(dY0|0)X^1|0−1. G(0,Y_0 0)=X_1 0^- (\! Y_0 0\! )X_1 0^-1- X_1 0^- \! (\! Y_0 0\! )\! X_1 0^-1. (23) Applying the vectorization (vecvec) operator to both sides of (23), and using the identity vec(PQR)=(R⊤⊗P)vec(Q)vec(PQR)=(R P)vec(Q), we get vecdG(0,Y0|0) vec\> G(0,Y_0 0) =[(X1|0−⊤⊗X1|0−⊤)−(X^1|0−⊤⊗X^1|0−⊤)]vecdY0|0. = [ (X_1 0^- X_1 0^- )- ( X_1 0^- X_1 0^- ) ]vec\> Y_0 0. (24) From (24), using the Jacobian identification rule [13, p. 199], we arrive at (18). ∎ Remark 2. A tedious computation shows that at θ≠0θ≠ 0, DY0|θG= D_Y_0 θG= X1|θ−⊤⊗X1|θ−⊤−X^1|θ−⊤⊗X^1|θ−⊤ X_1 θ^- X_1 θ^- - X_1 θ^- X_1 θ^- +θ∫01(Us|θ⊗Vs|θ+Vs|θ⊗Us|θ)ds, +θ _0^1 (U_s θ V_s θ+V_s θ U_s θ ) s, (25) where Us|θ U_s θ :=X^1|θ−⊤X^s|θ⊤Xs|θ−⊤, := X_1 θ^- X_s θ X_s θ^- , (26a) Vs|θ V_s θ :=X^1|θ−⊤X^s|θ⊤Πs|θBsRs−1Bs⊤Xs|θ−⊤. := X_1 θ^- X_s θ _s θB_sR_s^-1B_s X_s θ^- . (26b) While (25) generalizes (18), we will not use this result in the proofs that follow. Proposition 3. The partial Jacobian DY0|0GD_Y_0 0G evaluated at Y0|0−Y_0 0^- in (17), is nonsingular. Proof. For convenience, let Ψ01:= _01:= Σ0−12(14I+Σ012(Φ0→112)−1Σ1(Φ0→112)−⊤Σ012)12Σ0−12, _0^- 12\! (\! 14I+ _0 12\! (\! ^12_0→ 1\! )^-1 _1 (\! ^12_0→ 1\! )^- _0 12\! )^\! 12\! _0^- 12, (27a) S0:=12Σ0−1. S_0:= 12 _0^-1. (27b) As explained in Sec. I-B, X^0|θ=X0|θ=I X_0 θ=X_0 θ=I ∀θ∈[0,1]∀θ∈[0,1], so the matrices X^1|0=Φ0→1|011+Φ0→1|012Y^0|0− X_1 0= ^11_0→ 1 0+ ^12_0→ 1 0 Y_0 0^- and X1|0=Φ0→1|011+Φ0→1|012Y0|0−X_1 0= ^11_0→ 1 0+ ^12_0→ 1 0Y_0 0^- simplify to X^1|0 X_1 0 =X1|0−Φ0→1|012Σ0−1=−Φ0→1|012(Ψ01+S0), =X_1 0- ^12_0→ 1 0 _0^-1=- ^12_0→ 1 0\! ( _01+S_0 ), (28a) X1|0 X_1 0 =−Φ0→1|012(Ψ01−S0), =- ^12_0→ 1 0 ( _01-S_0 ), (28b) where we have used (13) and (17). Per Assumption A3, both Ψ01,S0 _01,S_0 in (27) are strictly positive definite. In particular, the positive definiteness of Ψ01 _01 follows from that • a positive definite matrix has unique (principal) positive definite square root, • Φ0→112 ^12_0→ 1 is invertible [7, Lemma 3], • the matrix Σ012(Φ0→112)−1Σ1(Φ0→112)−⊤Σ012≻0 _0 12\! (\! ^12_0→ 1\! )^-1 _1 (\! ^12_0→ 1\! )^- _0 12 0 since it is congruent [14, Ch. 4.5] to Σ1≻0 _1 0 via congruence transform defined by invertible matrix Σ012(Φ0→112)−1 _0 12\! (\! ^12_0→ 1 )^\!-1. Hence, Ψ01+S0≻0 _01+S_0 0. Furthermore, λi(14I+Σ012(Φ0→112)−1Σ1(Φ0→112)−⊤Σ012)>14∀i∈⟦n⟧ _i\! (\! 14I+ _0 12\! (\! ^12_0→ 1\! )^-1 _1 (\! ^12_0→ 1\! )^- _0 12\! )\!> 14 ∀ i∈ n implies that Ψ01−S0= _01-S_0= Σ0−12((14I+Σ012(Φ0→112)−1Σ1(Φ0→112)−⊤Σ012)12−12I)Σ0−12 _0^- 12\!\! (\!\! (\! 14I+ _0 12\!\! (\! ^12_0→ 1\! )^-1 _1 (\! ^12_0→ 1\! )^- _0 12\! )^\! 12\!\!- 12I\! )\! _0^- 12 ≻0. 0. (29) From (28), taking the inverse transposes, we get X^1|0−⊤ X_1 0^- =−(Φ0→1|012)−⊤(Ψ01+S0)−1, =- ( ^12_0→ 1 0 )^- \! ( _01+S_0 )^-1, (30a) X1|0−⊤ X_1 0^- =−(Φ0→1|012)−⊤(Ψ01−S0)−1. =- ( ^12_0→ 1 0 )^- \! ( _01-S_0 )^-1. (30b) Substituting (30) in (18) from Lemma 1, and using Kronecker product properties, we obtain DY0|0G|Y0|0=Y0|0−=((Φ0→112)−⊤⊗(Φ0→112)−⊤)Ω, D_Y_0 0G |_Y_0 0=Y_0 0^-=\! (\! ( ^12_0→ 1 )^- \! \! ( ^12_0→ 1 )^- ) , (31) where Ω:= := ((Ψ01−S0)−1⊗(Ψ01−S0)−1) (( _01-S_0)^-1 ( _01-S_0)^-1 ) −((Ψ01+S0)−1⊗(Ψ01+S0)−1). - (( _01+S_0)^-1 ( _01+S_0)^-1 ). (32) Since Φ0→112 ^12_0→ 1 is invertible, (31) is nonsingular if and only if detΩ≠0det\> ≠ 0. Since S0≻0S_0 0 and Ψ01 _01 are real symmetric, by the simultaneous diagonalizability theorem [14, Theorem 7.6.4(a)], [15, Theorem 20.1], there exists nonsingular P such that P⊤S0P=I,P⊤Ψ01P=diag(ψ1,…,ψn), P S_0P=I, P _01P=diag( _1,…, _n), (33) where ψi>0 _i>0 ∀i∈⟦n⟧∀ i∈ n denote the iith eigenvalue of Ψ01≻0 _01 0. Since Ψ01−S0≻0 _01-S_0 0 (see (29)), the simultaneous diagonalizability (33) implies the stronger: ψi>1 _i>1 ∀i∈1,…,n∀ i∈\1,…,n\. Then the eigenvalues of (Ψ01−S0)−1 ( _01-S_0 )^-1 and (Ψ01+S0)−1 ( _01+S_0 )^-1 are 1ψi−1 1 _i-1 and 1ψi+1 1 _i+1, respectively. So the spectrum of Ω in (32) comprises of 1(ψi−1)(ψj−1)−1(ψi+1)(ψj+1) 1( _i-1)( _j-1)- 1( _i+1)( _j+1) = = 2(ψi+ψj)(ψi2−1)(ψj2−1)∀(i,j)∈⟦n⟧×⟦n⟧, 2( _i+ _j)( _i^2-1)( _j^2-1) ∀(i,j)∈ n × n , which are positive because ψi>1 _i>1 ∀i∈⟦n⟧∀ i∈ n . This proves Ω≻0 0, and in particular detΩ≠0det\> ≠ 0. Hence (31) is nonsingular. ∎ Proof of Theorem 1. Specializing Proposition 2 at Y0|0=Y0|0−Y_0 0=Y_0 0^-, the mapping G is 1C^1 in a neighborhood of (0,Y0|0−)(0,Y_0 0^-). Moreover, since Y0|0−Y_0 0^- is the unique admissible risk-neutral solution, from (16), we have that G(0,Y0|0−)=0G(0,Y_0 0^-)=0. By Proposition 3, the partial Jacobian DY0|0GD_Y_0 0G evaluated at Y0|0−Y_0 0^-, is nonsingular. Hence, the implicit function theorem [16, Theorem 3.4] applies at (0,Y0|0−) (0,Y_0 0^- ). Therefore, there exist an open interval ℐ⊂ℝI containing θ=0θ=0, a neighborhood ⊂nU _n containing Y0|0Y_0 0, and a unique 1C^1 mapping h:ℐ→h:I such that h(0)=Y0|0−h(0)=Y_0 0^- and G(θ,h(θ))=0G(θ,h(θ))=0 ∀θ∈ℐ∀θ . This completes the proof. ∎ IV Numerical Example As an illustrative example, we implemented the solution of the LEQG covariance steering over a noisy double integrator, i.e., an instance of problem (2) with (At,Bt)≡([0100],[01]),Bt≡Ft,Qt=Rt=I∀t∈[0,1],(A_t,B_t)\!≡\! (\! bmatrix0&1\\ 0&0 bmatrix\!\!,\!\! bmatrix0\\ 1 bmatrix\! )\!,B_t≡ F_t,Q_t=R_t=I\>∀ t∈[0,1], and randomly generated endpoint covariances Σ0=[3.3356−0.0796−0.07962.0147],Σ1=[6.9230−2.0741−2.07413.7050]. _0=\! bmatrix3.3356&-0.0796\\ -0.0796&2.0147 bmatrix\!, _1=\! bmatrix6.9230&-2.0741\\ -2.0741&3.7050 bmatrix\!. These problem data satisfy assumptions A1-A3. With these problem data, we used Newton’s method with backtracking line search to solve the implicit nonlinear equation (16) in unknown Y0|θY_0 θ. Starting with Y0|0−Y_0 0^- in (17) as an initial guess for Y0|θY_0 θ, we solved the IVP defined by (9b) and Π0|θ=Y0|θ _0 θ=Y_0 θ, followed by the IVP defined by (9a) and H0|θ=Σ0−1−Π0|θH_0 θ= _0^-1- _0 θ. Due to the one-way coupling among (9a)-(9b), the latter IVP solution uses the trajectory Πt|θ _t θ from the former IVP. For the Newton update, we used the Jacobian (25). Fig. 1 shows the steering of the optimally controlled state covariance Σt|θ _t θ for θ=−1,0,1θ=-1,0,1. Fig. 1: LEQG steering of the state covariance Σt|θ _t θ, shown as respective 2σ2σ ellipses for t∈[0,1]t∈[0,1], θ∈−1,0,1θ∈\-1,0,1\, and state domain [−6,6]×[−4,4][-6,6]×[-4,4]. All ellipses are centered at the origin. As θ increases from negative to positive, the optimal solution transitions from risk-seeking to risk-averse. Consequently, logdetΣt|θ det _t θ for any fixed t, is a decreasing function of θ. Fig. 2 shows this hedging against uncertainty for the optimal solution as θ increases from −1-1 to +1+1 with step-size 0.10.1. Fig. 2: Risk-sensitivity parameter (θ) versus the log-determinant of the optimally controlled state covariance for t=0.3,0.6,0.9t=0.3,0.6,0.9. Fig. 3 also captures the same in that the optimal cost is shown to be increasing w.r.t. θ. Notice in particular that beyond the apparent scaling, the optimal 1θlog[eθC] 1θ [e^θ C] depends on θ because the random variable C in (1) depends on Πt|θ _t θ via (4), and also because the [⋅]E [· ] therein is taken w.r.t. the controlled joint state distribution (0,Σt|θ)N (0, _t θ ). Fig. 3: Risk-sensitivity parameter (θ) versus the optimal cost (2a). V Concluding Remarks The purpose of this letter was to introduce the LEQG covariance steering problem and to explain how its solution generalizes the results known in the literature, emphasizing new structural aspects of the solution. 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