Bi-Objective Optimal Control

• Bi-objective optimal control is a framework where controllers optimize two conflicting objectives under dynamic constraints using Pareto and Nash principles.
• The formulation employs scalarization techniques and variational methods to characterize Pareto fronts and establish existence conditions.
• Computational algorithms such as weighted sum, Chebyshev scalarization, and bisection methods enable efficient solutions across diverse system classes.

A bi-objective optimal control problem is a class of decision problem wherein the controller must simultaneously optimize two (typically conflicting) objective functionals subject to given system dynamics and constraints. These arise in diverse contexts including distributed parameter systems, stochastic processes, and multiagent games. Recent research has established rigorous frameworks for definition, existence and computation of Pareto equilibria and Nash strategies, as well as algorithmic and numerical methods for tracing the Pareto front or solving bi-objective lexicographic Markov Decision Processes (MDPs).

1. Mathematical Formulation and Pareto Optimality

The fundamental setting involves a controlled dynamical system (finite or infinite-dimensional, deterministic or stochastic), specified by state dynamics (e.g., ODE, PDE, Markov process), constrained admissible controls, and two objective functionals $J_1$, $J_2$ to be minimized (or one minimized, one maximized) over all admissible controls (and, in distributed cases, possibly over initial-final times or spatial domains):

$\min_{u(\cdot)} \; (J_1[u],\ J_2[u])$

with

$$J_i[u] = \int_{0}^T \ell_i(x(t),u(t),t)\,dt + \varphi_i(x(T)), \quad i=1,2$$

or, for Markov/stochastic systems, expectations over trajectories.

A control $u^*$ is Pareto optimal if there is no admissible $u$ with $J_i[u] \leq J_i[u^*]$ for all $i$ and strict inequality in at least one. The full solution set is the Pareto set; its image in $(J_1,J_2)$ space is the Pareto front. Weak Pareto optimality (strictly less in all components) is also standard (Kaya et al., 2023, Carvalho et al., 2023, Girejko et al., 2010).

Alternative formulations, motivated by applications in games and multiobjective synthesis, include Nash equilibria, where each control variable is partitioned among several agents, each optimizing one criterion, leading to a coupled fixed-point system (Buda et al., 8 Dec 2025, Buda et al., 14 Dec 2025).

2. Structural Results: Existence, Characterization, and Optimality Conditions

Existence of Pareto equilibria or efficient points in bi-objective optimal control depends strongly on problem convexity. For convex (linear quadratic, linear PDE, or convex semilinear) settings, classical results apply: Gâteaux or Fréchet differentiability, convexity of combined cost, and compactness/coercivity of the admissible set yield existence and structure of Pareto fronts and Nash equilibria. In the unconstrained $\mu > 0$ quadratic setting, the entire Pareto set is parameterized by a scalar weight $\alpha \in [0,1]$ (Carvalho et al., 2023).

A general characterization is as follows:

• For each $\alpha \in (0,1)$, define the scalarized objective $J_{(\alpha)}[u]=\alpha J_1[u] + (1-\alpha)J_2[u]$. Under convexity, every minimizer of $J_{(\alpha)}$ is Pareto efficient, and the whole front is generated by varying $\alpha$ (Carvalho et al., 2023, Kaya et al., 2023, Gattami, 2014, Girejko et al., 2010).
• For nonconvex or time-delayed systems, subregions of the Pareto front can be traced by Chebyshev scalarization. The set of attainable objective values is connected and, under mild regularity, locally smooth (Kaya et al., 2023).
• Necessary and sufficient optimality conditions are given by Pontryagin-type or generalized Euler–Lagrange systems augmented with vector-valued, Lagrange-multiplier weighted Hamiltonians, or by variational inequalities in the Nash game setting (Buda et al., 8 Dec 2025, Carvalho et al., 2023, Girejko et al., 2010).

In time-scale calculus and discrete/continuous hybrids, the delta–nabla unified framework extends these conditions to arbitrary time scales, showing Pareto points are extremal for weighted-sum/isoperimetric scalarizations (Girejko et al., 2010).

3. Representative System Classes and Model-Specific Structures

Bi-objective optimal control has been analyzed in a wide range of system classes:

• Diffusive PDEs: Linear, semilinear, bilinear (multiplicative control) heat equations in bounded domains, with objective functionals for $L^2$ tracking of multiple terminal targets and penalization of control effort (Carvalho et al., 2023).
• Stochastic MDPs: Lexicographically ordered objectives, e.g., maximize reachability then minimize conditional expected steps; maximize safety then conditional mean-payoff (Busatto-Gaston et al., 2023).
• Fractional PDEs: Space-time fractional diffusion with Caputo time- and space-fractional Laplacians, Nash games over distributed controls towards separate targets (Buda et al., 8 Dec 2025).
• Stokes equations: Velocity-pressure system, distributed controls, $L^2$-norm constraints, Nash equilibrium between agents with distinct target states. Finite element analysis yields optimal error bounds (Buda et al., 14 Dec 2025).
• Nonlinear dynamical systems: Bi-objective theory applied to nonlinear switched circuits (boost converters), with explicit analytic Pareto front for stability vs. power extraction (Numata et al., 22 Sep 2025).
• Stochastic LQG systems: Bi-objective LQG with energy constraints, solved via Riccati recursion with Lagrangian duality and bisection line search (Lee et al., 2021).
• SBM noise: Discrete systems with additive and multiplicative noises, quadratically-valued costs (possibly indefinite), SDP-based Pareto frontier and linear/affine optimal controllers (Gattami, 2014).

4. Computational Algorithms and Pareto Front Tracing

Algorithmic approaches fall into several principal classes:

• Scalarization techniques: Weighted sum, Chebyshev, and isoperimetric scalarization reduce the multiobjective problem to a family of single-objective (parameterized by scalar weight) control problems (Carvalho et al., 2023, Kaya et al., 2023, Girejko et al., 2010, Gattami, 2014).
  • Chebyshev scalarization is particularly effective in nonconvex and non-smooth settings, where the full front cannot be traced by convex combinations.
  • Essential interval of scalarization weights can be computed explicitly to bracket all non-degenerate Pareto points (Kaya et al., 2023).
• Bisection methods: For bilevel or constrained optimization over the Pareto set (e.g., minimizing a secondary cost on the Pareto front), a root-finding (bisection) procedure over the scalarization weight zeroes the derivative of the upper-level cost with respect to the scalarization parameter (Kaya et al., 2023, Lee et al., 2021).
• Variational inequality and gradient methods: In PDE or Nash game contexts, projected-gradient (e.g., conjugate gradient, gradient descent, Newton, fixed-point iteration) algorithms are applied directly to discrete optimality or variational inequality systems (Carvalho et al., 2023, Buda et al., 8 Dec 2025, Buda et al., 14 Dec 2025).
• Evolutionary and metaheuristic algorithms: For high-dimensional, nonconvex, or non-differentiable cases, evolutionary methods such as NSGA-II efficiently compute an empirical Pareto front from population-based search (Heris et al., 2020).
• SDP-based methods: Finite and infinite-horizon stochastic quadratic models with general quadratic cost criteria are reformulated as tractable SDPs in joint covariance variables; Pareto front recovery proceeds via parametric SDP solves (Gattami, 2014).

In MDPs with lexicographically related objectives, a two-stage procedure—first pruning suboptimal actions with respect to the primary criterion via Bellman backup or LP, then solving the secondary criterion in a reduced/pruned MDP—yields polynomial-time algorithms with correctness guarantees (Busatto-Gaston et al., 2023).

Method	System Class	Pareto Front Recovery
Weighted sum	PDE/ODE, stochastic	Vary scalarization parameter
Chebyshev/scalar	Nonconvex ODE/PDE	Essential weight interval
Two-stage pruning	MDP, lexicographic objs	Prune & re-solve
Variational ineq.	Nash (PDE), coupled obj.	Projected gradient methods
Evolutionary (NSGA-II)	Large/nonconvex, DICE	Nondominated sorting
SDP-based	LQ/Stochastic, quadratic	Solve parametric SDPs

5. Detailed Examples and Empirical Findings

Empirical and computational studies have validated these frameworks in diverse domains:

• Frozen Lake MDP: Implementing a two-stage lexicographic solution for reachability and steps-to-goal yields “sigma_DistOpt” which outperforms standard reachability-only strategies by large margins in path length, with identical reachability probabilities (e.g., up to $10^5\times$ shorter in hardest layouts) (Busatto-Gaston et al., 2023).
• Linear/Semilinear Heat Equations: The Pareto fronts are smooth and convex-like; parameter $\mu$ (control effort weight) controls the “shrinking” of the front toward the diagonal in objective space (Carvalho et al., 2023).
• Fractional Nash PDE: Conjugate gradient and FEM/L1 time stepping yield numerically efficient Nash solutions. Parametric studies confirm convergence in $L^2$ error rates as fractional order tends to one (Buda et al., 8 Dec 2025).
• Climate-Economy (DICE) Model: NSGA-II computes a Pareto front for welfare vs. peak temperature deviation. Results show that a negligible reduction in economic welfare can cut peak warming by over $1^{\circ}$C, while single-objective MPC lies suboptimal for temperature by comparison (Heris et al., 2020).
• Boost Converter: Complete analytic expressions for both objectives allow zero-approximation tracing of the trade-off; dimensionless parameters explicitly shape the Pareto contour (Numata et al., 22 Sep 2025).
• LQG/SDP: Riccati+bisection yields vastly more efficient Pareto designs than general-purpose SDP solvers on large horizons (e.g., $10\times$ faster for $N=1000$ steps) (Lee et al., 2021).

6. Theoretical Extensions and Generalizations

Recent research has clarified that the structure of the Pareto set depends not only on convexity but on the ability to “condition” secondary objectives on events achieving the primary criterion. In lexicographic MDPs, any two objectives where the secondary is “related” to the first by conditionalization (e.g., by reweighting transitions by value ratios) admit to similar two-stage pruning and resolving (Busatto-Gaston et al., 2023). Such constructions extend to chains of more than two objectives and cover parity/mean payoff and similar hybrid criteria.

Nash equilibrium analyses for PDEs and fractional systems have established existence and uniqueness by showing strict convexity and coercivity of each agent’s cost functional, leading to contraction mappings or variational inequalities whose solvability (often via Lions–Stampacchia-type arguments) guarantees unique equilibria (Buda et al., 8 Dec 2025, Buda et al., 14 Dec 2025).

Delta–nabla/time-scale optimality conditions demonstrate that the unified variational calculus encompasses continuous, discrete, and hybrid time flows, with direct sufficiency/necessity links between weighted-sum solutions and isoperimetric formulations (Girejko et al., 2010).

7. Significance, Applications, and Outlook

Bi-objective optimal control is central to modern control, operations research, and multiagent systems, particularly where competing design requirements or agent objectives demand explicit trade-off navigation. The rigorous mathematical characterizations of Pareto and Nash equilibria, combined with computationally efficient algorithms (scalarization, two-stage pruning, SDP reformulation, variational methods, evolutionary search) provide practical solutions for high-dimensional, constrained, and even nonconvex problems.

Emerging directions include expanding algorithmic tractability in high-dimensional, nonconvex, or uncertain environments, multi-level/multi-agent extensions (beyond two objectives), and robustification against modeling and stochastic uncertainties. The systematic identification of relationships between objectives that permit efficient pruning or reduction remains an active area of foundational and applied research.

References:

(Carvalho et al., 2023, Busatto-Gaston et al., 2023, Kaya et al., 2023, Buda et al., 8 Dec 2025, Buda et al., 14 Dec 2025, Lee et al., 2021, Heris et al., 2020, Numata et al., 22 Sep 2025, Gattami, 2014, Girejko et al., 2010)